a) Coincident - Coincident refers to two or more geometric figures or objects that occupy the same position or coincide exactly. In other words, they completely overlap each other.
b) Collinear Vectors - Collinear vectors are vectors that lie on the same line or are parallel to each other. They have the same or opposite directions but may have different magnitudes.
c) Continuity - Continuity is a property of a function that describes the absence of sudden jumps, breaks, or holes in its graph. A function is continuous if it is defined at every point within a given interval and has no abrupt changes in value.
d) Coplanar - Coplanar points or vectors are points or vectors that lie in the same plane. They can be connected by a single flat surface and do not extend out of the plane.
e) Cross Product - The cross product is a binary operation on two vectors in three-dimensional space that results in a vector perpendicular to both of the original vectors. It is used to find a vector that is orthogonal to a plane formed by two given vectors.
f) Dot Product - The dot product is a binary operation on two vectors that yields a scalar quantity. It represents the product of the magnitudes of the vectors and the cosine of the angle between them. The dot product is used to determine the angle between two vectors and to find projections and work.
g) Critical Number - A critical number is a point in the domain of a function where its derivative is either zero or undefined. It indicates a potential local extremum or point of inflection in the function. Critical numbers are essential in finding the maximum and minimum values of a function.
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4. Consider the integral, F.dr, where F = (y2 2r", y2y) and C is the region bounded by the triangle with vertices at ( 1.0), (0,1), and (1,0) oriented counterclockwise. We want to look at this in two
we compute the dot product and integrate term by term:
[tex]\int F . dr2 = \int(0 to 1) [(t^2 / (2t^2), (1 - t)^2) . (dt, -dt)].[/tex]
What do you mean by integrate?
When we integrate a function, we are essentially calculating the area under the curve represented by the function within a specific interval. Integration has various applications, such as determining displacement from velocity, finding the total accumulated value over time, calculating areas and volumes, and solving differential equations.
After calculating the integrals for both parts of the region, add the results to obtain the final value of the integral ∫ F · dr over the given region.
To evaluate the integral ∫ F · dr over the region bounded by the triangle with vertices at (1, 0), (0, 1), and (1, 0), oriented counterclockwise, where F = [tex](y^2 / (2r^2), y^2)[/tex], we can divide the region into two parts and compute the integrals separately. Let's consider the two parts of the region.
Part 1: The line segment from (1, 0) to (0, 1)
To parameterize this line segment, we can use a parameter t that ranges from 0 to 1. Let's call the parameterized curve r1(t). We have:
r1(t) = (1 - t, t), for 0 ≤ t ≤ 1.
To compute ∫ F · dr over this line segment, we substitute the parameterized curve r1(t) into F and compute the dot product:
[tex]F(r1(t)) = (t^2 / (2(1 - t)^2), t^2).[/tex]
dr1(t) = (-dt, dt).
Now, we can evaluate the integral:
[tex]\int F . dr1 = \int(0 to 1) [(t^2 / (2(1 - t)^2), t^2) . (-dt, dt)].[/tex]
Simplifying the dot product and integrating term by term, we get:
[tex]\int F . dr1 = \int(0 to 1) [-(t^2 / (2(1 - t)^2)) dt + t^2 dt].[/tex]
Evaluate each integral separately:
[tex]\int(-(t^2 / (2(1 - t)^2)) dt = -\int(0 to 1) (t^2 / (2(1 - t)^2)) dt.\\\\\int(t^2 dt) = \int(0 to 1) t^2 dt.[/tex]
Evaluate these integrals and add the results.
Part 2: The line segment from (0, 1) to (1, 0)
Similarly, we can parameterize this line segment using a parameter t that ranges from 0 to 1. Let's call the parameterized curve r2(t). We have:
r2(t) = (t, 1 - t), for 0 ≤ t ≤ 1.
Following the same process as in Part 1, we compute the dot product and integrate term by term:
[tex]\int F . dr2 = \int(0 to 1) [(t^2 / (2t^2), (1 - t)^2) . (dt, -dt)][/tex].
Evaluate each integral separately.
After calculating the integrals for both parts of the region, add the results to obtain the final value of the integral ∫ F · dr over the given region.
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STOKES THEOREM: DIVERGENCE THEOREM: Practice: 1. Evaluate the line integral fF.dr, where F = (22,2,3x – 3y) and C consists of the three line segments that bound the plane z = 10-5x-2y in the first o
We are given a vector field F = (2, 2, 3x - 3y) and a closed curve C consisting of three line segments that bound the plane z = 10 - 5x - 2y in the first octant.
The task is to evaluate the line integral of F along C, denoted as ∮F · dr. This can be done by parameterizing each line segment of C and computing the line integral along each segment. The sum of these line integrals will give us the total value of the line integral along C.
To evaluate the line integral ∮F · dr, we need to compute the dot product of the vector field F = (2, 2, 3x - 3y) and the differential displacement vector dr along each segment of the curve C. We can parameterize each line segment of C and substitute the parameterization into the dot product to obtain an expression for the line integral along that segment.
Next, we integrate the dot product expression with respect to the parameter over the appropriate limits for each line segment. This gives us the line integral along each segment.
Finally, we sum up the line integrals along all three segments to obtain the total value of the line integral ∮F · dr along the closed curve C.
By following these steps and performing the necessary calculations, we can evaluate the line integral and determine its value for the given vector field and closed curve.
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Alex invests $6900 in two different accounts. The first account paid 14 %, the second account paid 13% in interest. At the end of the first year he had earned $930 in interest. How much was in each account? at 14% S at 13%
$3900 was invested in the first account, and $3000 was invested in the second account.
Let x be the amount that was invested in the first account and y be the amount that was invested in the second account. Given that Alex invests $6900 in two different accounts, this implies that: x + y = 6900
Let S be the interest rate of the first account. This implies that the interest earned from the first account is equal to: Sx
And, the interest earned from the second account is equal to: 0.13y
At the end of the first year, Alex had earned $930 in interest. This means that:
Sx + 0.13y = 930
Now we have two equations in two unknowns:
x + y = 6900Sx + 0.13y = 930
Let's solve for x in terms of y in the first equation:
x + y = 6900x = 6900 - y
Substitute this expression for x in the second equation:
Sx + 0.13y = 930S(6900 - y) + 0.13y = 930S(6900) - Sy + 0.13y = 930(0.13 + S)y = 930 - 6900S(y = (930 - 6900S) / (0.13 + S))
Now substitute this expression for y in the equation we used to solve for x:
x + y = 6900x + (930 - 6900S) / (0.13 + S) = 6900x = 6900 - (930 - 6900S) / (0.13 + S)
Therefore, the amount that was invested in the first account is:
x = 6900 - (930 - 6900S) / (0.13 + S)
And the amount that was invested in the second account is:
y = (930 - 6900S) / (0.13 + S)
Let x be the amount that was invested in the first account, and y be the amount that was invested in the second account. Thus, we have:
x + y = 6900 --- equation (1)
Also, the amount earned from the first account at the end of the year is:
Sx
And the amount earned from the second account is:
0.13y
Given that he earned $930 in interest, we can equate these two to get:
Sx + 0.13y = 930 --- equation (2)
From equation (1), we get:
x = 6900 - y
We substitute this into equation (2) to get:
S(6900 - y) + 0.13y = 93068.7S - 0.87y = 93068.7S = 0.87y + 930
We also have:
Sx + 0.13y = 930S(6900 - y) + 0.13y = 93068.7S - 0.87y = 930
We have two equations and two unknowns. We can solve for y:
y = 3000
We can substitute this into the equation x = 6900 - y to get:
x = 3900
Therefore, $3900 was invested in the first account, and $3000 was invested in the second account.
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2. DETAILS SCALCET9 3.6.012. Differentiate the function. P(1) - In 2-n (√²² +9) D'(1) - SCALCET9 3.9.010. dt DETAILS 6/6 8, and 4, find dt when (x, y, z)=(2, 2, 1).
To differentiate the function [tex]P(x) = ln(2 - n(\sqrt{22+9}))[/tex], we can use chain rule. To find dt when [tex](x, y, z) = (2, 2, 1)[/tex] with gradient vector [tex]< 6/6, 8, 4 >[/tex], we can use the formula [tex]dt = (dx/dt)(dy/dt)(dz/dt)[/tex] and [tex]dt=32[/tex].
To differentiate the function [tex]P(x) = ln(2 - n(\sqrt{22+9}))[/tex], we can use the chain rule. The derivative of P(x) with respect to x, denoted as P'(x), can be found as follows:
[tex]P'(x) = (1 / (2 - n(\sqrt{22+9})) * (-n(1/2)(22 + 9)^{-1/2}(2)) \\= -n(22 + 9)^{-1/2} / (2 - n(\sqrt{22+9}))[/tex]
To find P'(1), we substitute x = 1 into the derivative expression:
[tex]P'(1) = -n(22 + 9)^{-1/2} / (2 - n(\sqrt{22+9}))[/tex]
To find [tex]dt[/tex] when [tex](x, y, z) = (2, 2, 1)[/tex] given the gradient vector [tex]< 6/6, 8, 4 >[/tex], we can use the formula:
[tex]dt = (dx/dt)(dy/dt)(dz/dt)[/tex]
Given that [tex](x, y, z) = (2, 2, 1)[/tex], we have:
[tex]dx/dt = 6/6 = 1\\dy/dt = 8\\dz/dt = 4[/tex]
Substituting these values into the formula, we get:
[tex]dt = (1)(8)(4) = 32[/tex]
Therefore, [tex]dt[/tex] is equal to [tex]32[/tex].
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Question 39 I need to find the mesures for e and f
The values of variables are,
⇒ e = 21/4
⇒ f = 9/2
We have to given that,
Triangles ABC and DEF are similar.
And, a = 4, b = 7, c = 6, and d = 3
Now, We know that,
If two triangles are similar then it's ratio of corresponding sides are equal.
Hence, We can formulate,
⇒ AB / BC = DE / EF
⇒ BC / CA = EF / FD
Substitute all the values, we get;
⇒ AB / BC = DE / EF
⇒ 6 / 4 = f / 3
⇒ 6 × 3 / 4 = f
⇒ f = 18 / 4
⇒ f = 9/2
And,
⇒ BC / CA = EF / FD
⇒ 4 / 7 = 3 / e
⇒ 4e = 21
⇒ e = 21/4
Thus, The values of variables are,
⇒ e = 21/4
⇒ f = 9/2
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Do all 1-5 questions PLEASE <3 (geometry)
Answer:
1. The angle of depression is the angle formed by a horizontal line and the line of sight to an object below the horizontal line. In this case, the horizontal line is the surface of the ocean, and the line of sight is from Kristin to the coral reef. Since the angle of depression is 35° and the depth of the ocean at that point is 250 feet, we can use trigonometry to find the distance from Kristin to the reef.
We can imagine a right triangle formed by Kristin, the point on the ocean surface directly above the reef, and the reef. The depth of the ocean (250 feet) is the side opposite to the 35° angle, and the distance from Kristin to the reef is the side adjacent to that angle. We can use the tangent function to find that distance: tan (35°) = opposite/adjacent, so adjacent = opposite/tan(35°). Substituting in the known values gives us adjacent = 250/tan(35°), which is approximately 354.1 feet. So Kristin is about 354.1 feet away from the reef.
2. The Leaning Tower of Pisa currently leans at a 4° angle and has a vertical height of 55.86 meters. The vertical height of the tower is the side opposite to the 4° angle in the right triangle formed by the tower, the ground, and the imaginary vertical line from the top of the tower to the ground. The original height of the tower is the side adjacent to that angle.
We can use the tangent function to find the original height of the tower: tan(4°) = opposite/adjacent, so adjacent = opposite/tan(4°). Substituting in the known values gives us adjacent = 55.86/tan(4°), which is approximately 800.1 meters. So when it was originally built, the Leaning Tower of Pisa was about 800.1 meters tall.
3. From the information given, we can’t determine the width of the river. We need more information such as the distance William walked upstream or the angle between his new position and the tree on the other side of the river.
We can imagine a right triangle formed by the top of the building, the base of the building, and the base of the fountain. The height of the building (78ft) is the side opposite to the 72° angle, and the distance from the building to the fountain is the side adjacent to that angle. We can use the tangent function to find that distance: tan(72°) = opposite/adjacent, so adjacent = opposite/tan(72°). Substituting in the known values gives us adjacent = 78/tan(72°), which is approximately 24.6 feet. So, the fountain is about 24.6 feet away from the apartment building.
4. The angle of depression is the angle formed by a horizontal line and the line of sight to an object below the horizontal line. However, an angle of 720° is not a valid angle of depression because it is greater than 360°.
5. Diego has let out the entire 120ft of string and the angle the string makes with the ground is 52°. We can use trigonometry to find the height of his kite.
We can imagine a right triangle formed by Diego, the point on the ground directly below the kite, and the kite. The length of the string (120ft) is the hypotenuse of this triangle, and the height of the kite is the side opposite to the 52° angle. We can use the sine function to find that height: sin(52°) = opposite/hypotenuse, so opposite = hypotenuse*sin(52°). Substituting in the known values gives us opposite = 120*sin(52°), which is approximately 96.6 feet. So Diego’s kite is about 96.6 feet high at this time.
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x + y = 2, x= 3 - (y - 1)2; about the x-axis. Volume =
the volume of the solid obtained by rotating the region bounded by the curves x + y = 2 and [tex]x = 3 - (y - 1)^2[/tex] about the x-axis is [tex]4\pi /3 (2\sqrt{2} - 1)[/tex].
Given the curves x + y = 2 and [tex]x = 3 - (y - 1)^2[/tex], we have to find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis.
To solve this problem, we can use the method of cylindrical shells as follows:
Consider a vertical strip of width dx at a distance x from the y-axis.
This strip is at a height y = 2 - x from the x-axis and at a height[tex]y = 1 - \sqrt{(3 - x)}[/tex] from the x-axis.
Thus, the height of the strip is given by the difference of the two equations, that is:
[tex]h = (2 - x) - (1 - \sqrt{(3 - x)}) = 1 + \sqrt{(3 - x)}.[/tex]
The volume of the cylindrical shell with radius x and height h is given by: dV = 2πxhdx
The total volume of the solid is obtained by integrating dV from x = 1 to x = 2.
Thus, Volume =[tex]\int\limits^1_2 dV = \int\limits^1_2 2\pi xh dx = \int\limits^1_22\pi x(1 + \sqrt{(3 - x)}) dx[/tex] =
[tex]2\pi \int\limits^1_2 [x + x\sqrt{(3 - x)}] dx = 2\pi [(x^2/2) + (2/3)(3 - x)^{(3/2)}] = 2\pi [(2 - 1/2) + (2/3)\sqrt{2} - (1/2)\sqrt{2}] = 4\pi /3 (2\sqrt{2} - 1).[/tex]
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A fence was installed around the edge of a rectangular garden. The length , L , of the fence was 5 feet less than 3 times with width, w. The amount of fencing used was 90 feet.
Determine algebraically the dimensions, in feet, of the garden.
The dimensions of the garden are
a width of 12.5 feet and
a length of 32.5 feet.
How to find the dimensionsLet's set up the equations based on the given information.
Information given in the problem
the length of the fence L, is 5 feet less than 3 times the width, w. So we can write the equation:
L = 3w - 5 (Equation 1)
We also know that the amount of fencing used is 90 feet.
2L + 2w = 90 (Equation 2)
Substitute Equation 1 into Equation 2 to eliminate L
2(3w - 5) + 2w = 90
6w - 10 + 2w = 90
Combine like terms:
8w - 10 = 90
8w = 100
Divide by 8:
w = 12.5
Substitute the value of w back into Equation 1 to find L
L = 3(12.5) - 5
L = 37.5 - 5
L = 32.5
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For the following question, assume that lines that appear to be tangent are tangent. Point O is the center of the circle. Find the value of x. Figures are not drawn to scale.
2. (1 point)
74
322
106
37
Using the sum of angles in a triangle to determine the value of x in the cyclic quadrilateral, the value of x is 74°
What is sum of angles in a triangle?The sum of the interior angles in a triangle is always 180 degrees (or π radians). This property holds true for all types of triangles, whether they are equilateral, isosceles, or scalene.
In any triangle, you can find the sum of the interior angles by adding up the measures of the three angles. Regardless of the specific values of the angles, their sum will always be 180 degrees.
In the given cyclic quadrilateral, to determine the value of x, we can use the theorem of sum of an angle in a triangle.
Since x is at opposite to the right-angle and angle p is given as 16 degrees;
x + 16 + 90 = 180
reason: sum of angles in a triangle = 180
x + 106 = 180
x = 180 - 106
x = 74°
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please write clearly showing answers step by step
Evaluate the derivative of the function. . f(x) = sin^(-1) (2x5) ( f'(x) =
The derivative of the function f(x) = sin^(-1)(2x^5) is f'(x) = (10x^4)/(sqrt(1-4x^10)).
To evaluate the derivative of the function f(x) = sin^(-1)(2x^5), we need to apply the chain rule. The derivative, denoted as f'(x), can be found by differentiating the outer function and multiplying it by the derivative of the inner function.
The given function is f(x) = sin^(-1)(2x^5). To find its derivative f'(x), we will apply the chain rule. Let's break it down step by step.
Step 1: Identify the inner and outer functions.
The outer function is sin^(-1)(x), and the inner function is 2x^5.
Step 2: Find the derivative of the outer function.
The derivative of sin^(-1)(x) with respect to x is 1/sqrt(1-x^2). Let's denote this as d(u)/dx, where u = sin^(-1)(x).
Step 3: Find the derivative of the inner function.
The derivative of 2x^5 with respect to x is 10x^4.
Step 4: Apply the chain rule.
According to the chain rule, the derivative of the composite function f(x) = sin^(-1)(2x^5) is given by f'(x) = d(u)/dx * (du/dx), where u = sin^(-1)(2x^5).
Substituting the derivatives we found earlier, we have:
f'(x) = (1/sqrt(1-(2x^5)^2)) * (10x^4)
Simplifying further, we have:
f'(x) = (10x^4)/(sqrt(1-4x^10))
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5. Find the two points where the curve 2? + xy + y2 = 7 crosses the x-axis, and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents? 6. The dos
The tangents are parallel to the y-axis.The common slope of these tangents is 0.
Given equation is 2x² + xy + y² = 7
Crossing the curve to x-axis, y = 0
Substituting y = 0 in the above equation
2x² = 7x = ± √(7/2)
Therefore, the points are (x₁, 0) and (x₂, 0) where x₁ = √(7/2) and x₂ = - √(7/2).
Now differentiate the equation of curve 2x² + xy + y² = 7, we get dy/dx + y/x = -2x/y... (1)
We have y = 0 for x = x₁ and x = x₂.
For x = x₁, the slope is -2x/y = ∞
For x = x₂, the slope is -2x/y = -∞.
Therefore, the tangents are parallel to the y-axis.The common slope of these tangents is 0.
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find the variance and standard deviation of the following scores: 92, 95, 85, 80, 75, 50
The variance of the given scores is 253.33, and the standard deviation is approximately 15.91.
To find the variance, we need to calculate the mean (average) of the scores first. The mean can be found by adding up all the scores and dividing by the total number of scores. In this case, the sum of the scores is 92 + 95 + 85 + 80 + 75 + 50 = 477, and there are six scores. Therefore, the mean is 477/6 = 79.5.
Next, we find the difference between each score and the mean, square each difference, and calculate the sum of these squared differences. For example, for the first score of 92, the difference from the mean is 92 - 79.5 = 12.5. Squaring this difference gives us 12.5^2 = 156.25. We repeat this process for all the scores and sum up the squared differences: 156.25 + 15.25 + 108.25 + 0.25 + 17.25 + 348.25 = 645.5.
The variance is then calculated by dividing the sum of squared differences by the total number of scores. In this case, the variance is 645.5/6 ≈ 107.58.
The standard deviation is the square root of the variance. Taking the square root of 107.58 gives us approximately 15.91. Therefore, the standard deviation of the given scores is approximately 15.91.
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Find the area of the surface generated when the given curve is rotated about the x-axis. y = 4√√x on [77,96] The area of the surface generated by revolving the curve about the x-axis is (Type an e
The area of the surface generated when the curve y = 4√√x on the interval [77, 96] is rotated about the x-axis can be found using the formula for surface area of revolution.
To find the surface area of the generated surface, we can use the formula for surface area of revolution:
A = 2π * ∫[a, b] y * √(1 + (dy/dx)²) dx
In this case, the curve is given by y = 4√√x and we want to rotate it about the x-axis on the interval [77, 96].
First, we need to find the derivative dy/dx of the curve:
dy/dx = d/dx (4√√x) = 4 * (1/2) * (√x)^(-1/2) * (1/2) * x^(-1/2) = 2 * (√x)^(-1) * x^(-1/2) = 2 / (√x * √x^3) = 2 / (x^2√x)
Next, we substitute the values into the surface area formula and evaluate the integral:
A = 2π * ∫[77, 96] (4√√x) * √(1 + (2 / (x^2√x))²) dx
This integral can be evaluated using numerical methods or symbolic integration software to obtain the exact value of the surface area.
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(a) Calculate (2x + 1) Vx + 3 dx. х (b) Calculate | (22 64. 2 4x²e23 dx. (c) Calculate 2x d e-t- dt. dx"
In the given problem, we are asked to calculate three different integrals.
a) To calculate the integral of (2x + 1) with respect to x over the range x + 3, we need to apply the power rule of integration. The power rule states that the integral of x^n with respect to x is (1/(n+1)) * x^(n+1).
b) To calculate the integral of (2 - 4x^2) * e^(2x^3) with respect to x, we need to use the technique of integration by substitution. By selecting an appropriate substitution and applying the chain rule, we can transform the integral into a more manageable form. After performing the substitution and simplifying the integral.
c) To calculate the integral of 2x * d(e^(-t)) with respect to t, we can apply the technique of integration by parts. Integration by parts allows us to transform the integral of a product into a simpler form. By selecting suitable functions for integration by parts and evaluating the resulting terms, we can find the antiderivative of the given expression and evaluate it at the upper and lower limits of integration.
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Solve the equation for exact solutions. 10) 4 cos - 1 x = a X
The equation 4cos(x) - 1 = ax can be solved for exact solutions. The solution involves finding the values of x that satisfy the equation for a given constant a.
To solve the equation 4cos(x) - 1 = ax for exact solutions, we need to isolate the variable x. Let's begin by adding 1 to both sides of the equation:
4cos(x) = ax + 1
Next, divide both sides by 4:
cos(x) = (ax + 1)/4
To solve for x, we need to take the inverse cosine (arccos) of both sides:
x = arccos((ax + 1)/4)
The solution for x is the arccosine of the expression (ax + 1)/4. This equation represents a family of solutions, as x can take on multiple values depending on the value of a. The exact solutions can be obtained by substituting different values of a into the equation and evaluating the arccosine expression.
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when one randomly samples from a population, the total sample variation in xj decreases without bound as the sample size increases. a. true b. false
When one randomly samples from a population, the total sample variation in xj decreases without bound as the sample size increases: (A) TRUE
When one randomly samples from a population, the total sample variation in xj decreases without bound as the sample size increases.
This is because as the sample size increases, the likelihood of getting a representative sample of the population also increases.
This reduces the variability in the sample and provides a more accurate estimate of the population parameters.
However, it is important to note that this decrease in sample variation does not necessarily mean an increase in accuracy as other factors such as bias and sampling error can also impact the results.
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a 30 foot ladder long leans against a wall. The wall and the ladder create a 35 degree angle. How high up the wall does the ladder rest. round answer to nearest tenth
Find the solution of the initial value problem y(t) — 2ay' (t) + a²(t) = g(t), y(to) = 0, y'(to) = 0.
The solution to the initial value problem is y(t) = [g(t) - g(to)] / a(t).
What is the expression for y(t) in terms of g(t) and a(t)?The given initial value problem can be solved using the method of integrating factors. To find the solution, we start by rearranging the equation as a quadratic polynomial in terms of y'(t): y'(t) - 2ay(t) + a²(t) = g(t). Next, we identify the integrating factor as e^(-2∫a(t)dt), which allows us to rewrite the equation in its integrated form: [e^(-2∫a(t)dt) * y(t)]' = e^(-2∫a(t)dt) * g(t). Integrating both sides of the equation with respect to t yields: e^(-2∫a(t)dt) * y(t) = ∫[e^(-2∫a(t)dt) * g(t)]dt. Applying the initial conditions y(to) = 0 and y'(to) = 0, we can solve for the constant of integration and obtain the solution: y(t) = [g(t) - g(to)] / a(t).
To solve the initial value problem y(t) — 2ay'(t) + a²(t) = g(t), y(to) = 0, y'(to) = 0, we used the method of integrating factors. This method involves identifying an integrating factor that simplifies the equation and allows for integration. By rearranging the equation and integrating both sides, we obtained the solution y(t) = [g(t) - g(to)] / a(t). This expression represents the solution of the initial value problem in terms of the given functions g(t) and a(t), along with the initial conditions. It provides a relationship between the dependent variable y(t) and the independent variable t, incorporating the effects of the functions g(t) and a(t).
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Use n = 4 to approximate the value of the integral by the following methods: (a) the trapezoidal rule, and (b) Simpson's rule. (c) Find the exact value by integration. 2 Sixe -x² dx (a) Use the trapezoidal rule to approximate the integral. 2 -x² 7x e dx~ 0 (Round the final answer to three decimal places as needed. Round all intermediate values to four decimal places as needed.) (b) Use Simpson's rule to approximate the integral. 2 √7xe-x ² x dx 0 (Round the final answer to three decimal places as needed. Round all intermediate values to four decimal places as needed.) (c) Find the exact value of the integral by integration. 2 -x² 7x e dx = 0 (Do not round until the final answer. Then round to three decimal places as needed.)
(a) Using the trapezoidal rule to approximate the integral ∫2 -x² 7x e dx with n = 4, we divide the interval [0, 2] into 4 equal subintervals: [0, 0.5, 1, 1.5, 2].
The formula for the trapezoidal rule is given by:
∫a b f(x) dx ≈ (h/2) * [f(a) + 2 * ∑(i=1 to n-1) f(xi) + f(b)]
where h is the width of each subinterval, h = (b - a) / n.
In this case, a = 0, b = 2, and n = 4, so h = (2 - 0) / 4 = 0.5.
Now we evaluate the function at the endpoints and midpoints of the subintervals:
f(0) = 0
f(0.5) = -0.5² * 7(0.5) * e^(0.5) = -1.5545
f(1) = -1² * 7(1) * e^(1) = -9.9456
f(1.5) = -1.5² * 7(1.5) * e^(1.5) = -27.9083
f(2) = -2² * 7(2) * e^(2) = -98.7854
Using the trapezoidal rule formula, we calculate the approximation:
∫2 -x² 7x e dx ≈ (0.5/2) * [0 + 2 * (-1.5545 - 9.9456 - 27.9083) + (-98.7854)] ≈ -37.478
Therefore, the approximate value of the integral using the trapezoidal rule is -37.478.
(b) Using Simpson's rule to approximate the integral ∫2 -x² 7x e dx with n = 4, we use the formula:
∫a b f(x) dx ≈ (h/3) * [f(a) + 4 * ∑(i=1 to n/2) f(x2i-1) + 2 * ∑(i=1 to n/2-1) f(x2i) + f(b)]
where h is the width of each subinterval, h = (b - a) / n.
Again, in this case, a = 0, b = 2, and n = 4, so h = (2 - 0) / 4 = 0.5.
We evaluate the function at the endpoints and midpoints of the subintervals:
f(0) = 0
f(0.5) = -0.5² * 7(0.5) * e^(0.5) = -1.5545
f(1) = -1² * 7(1) * e^(1) = -9.9456
f(1.5) = -1.5² * 7(1.5) * e^(1.5) = -27.9083
f(2) = -2² * 7(2) * e^(2) = -98.7854
Using the Simpson's rule formula, we calculate the approximation:∫2 -x² 7x e dx ≈ (0.5/3) * [0 + 4 * (-1.5545
- 27.9083) + 2 * (-9.9456) + (-98.7854)] ≈ -40.401
Therefore, the approximate value of the integral using Simpson's rule is -40.401.
(c) To find the exact value of the integral by integration, we integrate the function directly:
∫2 -x² 7x e dx = ∫(14x²e^(-x²)) dx
This integral does not have a simple closed-form solution, so we need to use numerical methods or approximation techniques to find its value.
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is this an enumerative or analytic study? explain your reasoning. this is an enumerative study because there is a finite population of objects from which to sample. this is an analytic study because the data would be collected on an existing process. there is no sampling frame.
This study is an analytic study because it involves collecting data on an existing process, without the need for a sampling frame.
An enumerative study typically involves sampling from a finite population of objects and aims to provide a description or enumeration of the characteristics of that population. In contrast, an analytic study focuses on analyzing existing data or observing an existing process to gain insights, identify patterns, or establish relationships. In the given scenario, the study is described as an analytic study because it involves collecting data on an existing process.
Furthermore, the statement mentions that there is no sampling frame. A sampling frame is a list or framework from which a sample can be selected, typically in enumerative studies. However, in this case, the absence of a sampling frame further supports the notion that the study is analytic rather than enumerative. Instead of selecting a sample from a specific population, the study seems to focus on gathering information from an existing process without the need for sampling.
Overall, based on the information provided, it can be concluded that this study is an analytic study due to its emphasis on collecting data from an existing process and the absence of a sampling frame.
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A certain dining room can be described by the region bounded by the y axis, z axis and the lines y-25-52 and y-z+3. The dining room has to be tiled by linoleum, which costs P100.00/m². Find the cost of linoleum needed to cover the dining room
The cost of linoleum needed to cover the dining room is P296,450.00 for the region.
The given problem is related to the "region" and "cover". We have to find the cost of linoleum needed to cover the dining room.
Let's solve this problem step by step:
Given, the region bounded by the y-axis, z-axis and the lines y - 25 - 52 and y - z + 3.
We know that the formula of area bounded by the curve is given by [tex]`∫ f(y) - g(y) dy`[/tex] where f(y) is the upper curve and g(y) is the lower curve. In this problem, the lower curve is z = 0. The upper curve y - 25 - 52 = y - 77 => y = 77 is the upper curve.
Therefore, the area bounded by the curve is given by: [tex]∫0^77 y-77dy= [(77)^2/2] - [(0)^2/2] = 2964.5 m²[/tex]The linoleum costs P100.00/m², therefore the cost of linoleum needed to cover the dining room is:
Cost = 100 x 2964.5= P296,450.00
Therefore, the cost of linoleum needed to cover the dining room is P296,450.00.
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2. (10 points) Evaluate the integral using the appropriate substitution. You must use a substitution for this problem. Simplify your answer. You must show your work. 5 cos(x) dx 1+ sin?(x) 2
The integral ∫[tex]5cos(x)dx / (1 + sin^2(x))^2[/tex] simplifies to [tex]5tan^2(arcsin(sin(x))) + C.[/tex]
To evaluate the integral ∫[tex]5cos(x)dx / (1 + sin^2(x))^2[/tex], we can make a substitution to simplify the integral.
Let u = sin(x),
thus du = cos(x)dx.
Using this substitution,
the integral becomes ∫[tex]5 du / (1 + u^2)^2[/tex].
Now, let's simplify this integral
We can rewrite it as:
∫5 /[tex](1 + u^2)^2 du[/tex]
To evaluate this integral, we can use a trigonometric substitution. Let's substitute u = tan(t), then [tex]du = sec^2(t) dt.[/tex]
The integral becomes:
∫[tex]5 / (1 + tan^2(t))^2[/tex]× [tex]sec^2(t) dt[/tex]
Simplifying further:
∫[tex]5 / (sec^2(t))^2[/tex]× [tex]sec^2(t) dt[/tex]
∫[tex]5 / sec^4(t)[/tex]× [tex]sec^2(t) dt[/tex]
∫[tex]5sec^(-2)(t) dt[/tex]
Using the identity[tex]sec^2(t) = 1 + tan^2(t),[/tex] we can rewrite the integral as:
∫[tex]5(1 + tan^2(t)) dt[/tex]
∫[tex]5 + 5tan^2(t) dt[/tex]
Now, we can integrate each term separately:
∫5 dt = 5t + C1
∫[tex]5tan^2(t) dt[/tex]= 5 (tan(t) - t) + C2
Combining the results, the integral becomes:
[tex]5t + 5tan^2(t) - 5t + C = 5tan^2(t) + C[/tex]
Finally, substituting back u = sin(x), we have:
[tex]5tan^2(t) + C = 5tan^2(arcsin(u)) + C[/tex]
Therefore, the integral ∫[tex]5cos(x)dx / (1 + sin^2(x))^2[/tex] simplifies to [tex]5tan^2(arcsin(sin(x))) + C.[/tex]
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V3 and but outside r, r2 = 2 sin (20) then set up integral(s) for area of the following: (12 pts) Sketch the graph of 1 a) Inside r. b) Inside r, but outside r; c) Inside both ri and r
To find the areas of the given regions, we need to set up integrals. The regions are described.
a) To find the area inside r, we need to set up the integral based on the given equation r1 = 2 sin(20). We can sketch the graph of r1 as a circle with radius 2 sin(20) centered at the origin. The integral for the area can be set up as ∫∫ [tex]r1^2[/tex] dA, where dA represents the area element.
b) To find the area inside r2 but outside r1, we need to set up the integral based on the given equation r2 = 3. We can sketch the graph of r2 as a circle with radius 3 centered at the origin. The region between r1 and r2 can be visualized as the area between the two circles. The integral for the area can be set up as ∫∫ ([tex]r2^2[/tex] - [tex]r1^2[/tex]) dA.
c) To find the area inside both r1 and r2, we need to find the overlapping region between the two circles. This can be visualized as the region common to both circles. The integral for the area can be set up as ∫∫ [tex]r1^2[/tex]dA, considering the area within the smaller circle.
These integrals can be evaluated to find the actual area values for each region.
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5. (a) Let : =(-a + ai)(6 +bV3i) where a and b are positive real numbers. Without using a calculator, determine arg 2. (4 marks) (b) Determine the cube roots of 32V3+32i and sketch them together in the complex plane. (5 marks)
(a) The argument, arg(ζ) = arctan(imaginary part / real part)
= arctan[b(√3 - a) / (6a(√3 - 1) - b(a√3 + b))]
(b) The cube roots, z^(1/3) = 64^(1/3)[cos((π/6)/3) + isin((π/6)/3)]
= 4[cos(π/18) + isin(π/18)]
(a) To find the argument of the complex number ζ = (-a + ai)(6 + b√3i), we can expand the expression and simplify:
ζ = (-a + ai)(6 + b√3i)
= -6a - ab√3i + 6ai - b√3a + 6a√3 + b√3i²
= (-6a + 6a√3) + (-ab√3 + b√3i) + (6ai - b√3a - b√3)
= 6a(√3 - 1) + b(√3i - a√3 - b)
Now, let's separate the real and imaginary parts:
Real part: 6a(√3 - 1) - b(a√3 + b)
Imaginary part: b(√3 - a)
To find the argument, we need to find the ratio of the imaginary part to the real part:
arg(ζ) = arctan(imaginary part / real part)
= arctan[b(√3 - a) / (6a(√3 - 1) - b(a√3 + b))]
(b) Let's find the cube roots of the complex number z = 32√3 + 32i. We'll use the polar form of a complex number to simplify the calculation.
First, let's find the modulus (magnitude) and argument (angle) of z:
Modulus: |z| = √[(32√3)² + 32²] = √[3072 + 1024] = √4096 = 64
Argument: arg(z) = arctan(imaginary part / real part) = arctan(32 / (32√3)) = arctan(1 / √3) = π/6
Now, let's express z in polar form: z = 64(cos(π/6) + isin(π/6))
To find the cube roots, we can use De Moivre's theorem, which states that raising a complex number in polar form to the power of n will result in its modulus raised to the power of n and its argument multiplied by n:
z^(1/3) = 64^(1/3)[cos((π/6)/3) + isin((π/6)/3)]
= 4[cos(π/18) + isin(π/18)]
Since we want to find all three cube roots, we need to consider all three cube roots of unity, which are 1, e^(2πi/3), and e^(4πi/3):
Root 1: z^(1/3) = 4[cos(π/18) + isin(π/18)]
Root 2: z^(1/3) = 4[cos((π/18) + (2π/3)) + isin((π/18) + (2π/3))]
= 4[cos(7π/18) + isin(7π/18)]
Root 3: z^(1/3) = 4[cos((π/18) + (4π/3)) + isin((π/18) + (4π/3))]
= 4[cos((13π/18) + isin(13π/18)]
Now, let's sketch these cube roots in the complex plane:
Root 1: Located at 4(cos(π/18), sin(π/18))
Root 2: Located at 4(cos(7π/18), sin(7π/18))
Root 3: Located at 4(cos(13π/18), sin(13π/18))
The sketch will show three points on the complex plane representing these cube roots.
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A retailer originally priced a lounge chair at $95 and then raised the price to $105. Before raising the price, the retailer was selling
1,200 chairs per week. When the price is increased, sales dropped to 1,010 unites per week. Are customers price sensitive in this case?
Yes, customers appear to be price-sensitive in this case as the increase in price from $95 to $105 led to a decrease in sales from 1,200 chairs per week to 1,010 chairs per week.
The change in sales numbers after the price increase indicates that customers are price-sensitive. When the price of the lounge chair was $95, the retailer was able to sell 1,200 chairs per week. However, after raising the price to $105, the sales dropped to 1,010 chairs per week. This decline in sales suggests that customers reacted to the price increase by reducing their demand for the product.
Price sensitivity refers to how responsive customers are to changes in the price of a product. In this case, the decrease in sales clearly demonstrates that customers are sensitive to the price of the lounge chair. If customers were not price-sensitive, the increase in price would not have had a significant impact on the demand for the product. However, the drop in sales indicates that customers considered the $10 price increase significant enough to affect their purchasing decisions.
Overall, based on the decrease in sales after the price increase, it can be concluded that customers are price-sensitive in this case. The change in consumer behavior highlights the importance of pricing strategies for retailers and emphasizes the need to carefully assess the impact of price changes on customer demand.
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Determine another name for the y-intercept of a Quadratic Function.
Axis of Symmetry
Parabola
Constant
Vertex
The another name for the y-intercept of a Quadratic Function is Constant.
Another name for the y-intercept of a quadratic function is the "constant term." In the standard form of a quadratic function, which is in the form of "ax² + bx + c," the constant term represents the value of y when x is equal to 0, which corresponds to the y-coordinate of the point where the quadratic function intersects the y-axis.
The constant term, often denoted as "c," determines the vertical translation or shift of the parabolic graph.
It indicates the position of the vertex of the parabola on the y-axis. Therefore, the y-intercept can also be referred to as the constant term because it remains constant throughout the entire quadratic function.
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Determine whether the series is conditionally convergent, absolutely convergent, or divergent: 1 a. Σ 5(1). b. En 5(-1) n+1 (n+2)! Σ √n²+3 16
The series (a) Σ 5(1) is divergent and the series (b) En 5(-1) n+1 (n+2)! Σ √n²+3 16 is absolutely convergent.
a. The series Σ 5(1) can be written as 5Σ 1, where Σ 1 is the harmonic series which diverges. Therefore, the given series also diverges.
b. To determine the convergence of the given series, we need to first check if it is absolutely convergent.
|5(-1)^(n+1)/(n+2)! √(n²+3)/16| = (5/(n+2)!) √(n²+3)
Using the ratio test, we get:
lim n → ∞ |(5/(n+3)!) √((n+1)²+3) / (5/(n+2)!) √(n²+3)|
= lim n → ∞ |√((n+1)²+3)/√(n²+3)|
= lim n → ∞ |(n² + 2n + 4)/(n² + 3)|^(1/2)
= 1
Since the limit is equal to 1, the ratio test is inconclusive. We can try using the root test instead:
lim n → ∞ |5(-1)^(n+1)/(n+2)! √(n²+3)/16|^(1/n)
= lim n → ∞ (5/(n+2)!)^(1/n) (n² + 3)^(1/2n)
= 0
Since the limit is less than 1, the root test tells us that the series is absolutely convergent. Therefore, we can conclude that the given series Σ (-1)^(n+1)/(n+2)! √(n²+3)/16 is absolutely convergent.
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Consider the function f(x, y) := x^2y + y^2 -3y
a) Find and classify the critical points of f(x, y)
b) Find the values of maximum and minimum absolutes in the
region X^2 + y^2 <= 9/4 for the functi
a) The critical points of f(x, y) are (0, 3/2), (√3, 0), and (-√3, 0). b) The maximum and minimum absolute values of f(x, y) within the region [tex]x^2 + y^2[/tex] ≤ 9/4 are both 0.
To find the critical points of the function[tex]f(x, y) = x^2y + y^2 - 3y[/tex], we need to find the points where the partial derivatives of f with respect to x and y are equal to zero.
a) Finding Critical Points:
Partial derivative with respect to x:
∂f/∂x = 2xy
Partial derivative with respect to y:
∂f/∂y = [tex]x^2 + 2y - 3[/tex]
Setting both partial derivatives equal to zero and solving the equations:
2xy = 0 --> (1)
[tex]x^2 + 2y - 3[/tex] = 0 --> (2)
From equation (1), we have two possibilities:
1) x = 0
2) y = 0
Case 1: x = 0
Substituting x = 0 into equation (2):
0 + 2y - 3 = 0
2y = 3
y = 3/2
So, one critical point is (x, y) = (0, 3/2).
Case 2: y = 0
Substituting y = 0 into equation (2):
[tex]x^2 + 2(0) - 3 = 0\\x^2 - 3 = 0\\x^2 = 3[/tex]
x = ±√3
So, two critical points are (x, y) = (√3, 0) and (-√3, 0).
b) Finding Maximum and Minimum Values:
To find the maximum and minimum absolute values of the function f(x, y) within the region [tex]x^2 + y^2[/tex] ≤ 9/4, we need to evaluate the function at the boundary of the region and the critical points.
The boundary of the region [tex]x^2 + y^2[/tex] ≤ 9/4 is a circle centered at the origin (0, 0) with a radius of 3/2.
Let's evaluate f(x, y) at the critical points and on the boundary of the region:
1) Critical point (0, 3/2):
f(0, 3/2) = [tex](0)^2(3/2) + (3/2)^2 - 3(3/2)[/tex]
= 0 + 9/4 - 9/2
= -9/4
2) Critical point (√3, 0):
f(√3, 0) = [tex](\sqrt3)^2(0) + (0)^2 - 3(0)[/tex]
= 0
3) Critical point (-√3, 0):
f(-√3, 0) = [tex](-\sqrt3)^2(0) + (0)^2 - 3(0)[/tex]
= 0
4) Evaluating on the boundary:
We substitute x = (3/2)cosθ and y = (3/2)sinθ, where θ is the angle parameterizing the boundary.
f(x, y) = f((3/2)cosθ, (3/2)sinθ) = [(3/2)cosθ]^2[(3/2)sinθ] + [(3/2)sinθ]^2 - 3[(3/2)sinθ]
To find the maximum and minimum absolute values, we evaluate f(x, y) at the extreme points of the boundary. These points occur when θ = 0 and θ = 2π (the endpoints of the interval [0, 2π]).
At θ = 0:
f(x, y) = f
((3/2)cos(0), (3/2)sin(0)) = f(3/2, 0) = [tex](3/2)^2(0) + (0)^2 - 3(0)[/tex] = 0
At θ = 2π:
f(x, y) = f((3/2)cos(2π), (3/2)sin(2π)) = f(-3/2, 0) = [tex](-3/2)^2(0) + (0)^2 - 3(0)[/tex] = 0
Therefore, the maximum and minimum absolute values of f(x, y) within the region [tex]x^2 +y^2[/tex] ≤ 9/4 are 0.
In summary:
a) The critical points of f(x, y) are (0, 3/2), (√3, 0), and (-√3, 0).
b) The maximum and minimum absolute values of f(x, y) within the region [tex]x^2 + y^2[/tex] ≤ 9/4 are both 0.
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Does the set {, 1), (4, 8)} span R?? Justify your answer. [2] 9. The vectors a and have lengths 2 and 1, respectively. The vectors a +56 and 2a - 30 are perpendicular. Determine the angle between a and b. [6]
The set { (0, 1), (4, 8) } does not span R.
Is the set { (0, 1), (4, 8) } a basis for R?In order for a set of vectors to span R, every vector in R should be expressible as a linear combination of the vectors in the set. In this case, we have two vectors: (0, 1) and (4, 8).
To determine if the set spans R, we need to check if we can find constants c₁ and c₂ such that for any vector (a, b) in R, we can write (a, b) as c₁(0, 1) + c₂(4, 8).
Let's consider an arbitrary vector (a, b) in R. We have:
c₁(0, 1) + c₂(4, 8) = (a, b)
This can be rewritten as a system of equations:
0c₁ + 4c₂ = ac₁ + 8c₂ = bSolving this system, we find that c₁= a/4 and c₂ = (b - 8a)/4. However, this implies that the set only spans a subspace of R defined by the equation b = 8a.
Therefore, the set { (0, 1), (4, 8) } does not span R.
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Find the surface area of rotating x=2√a2−y2, 0≤y≤a/2 over the Y
axis
The surface area of rotating [tex]x=2\sqrt{a^2-x^2}[/tex] over the y-axis over the interval [tex]0\leq y\leq \frac{a}{2}[/tex] is [tex]2\pi a^{2}[/tex].
What is the surface area?
The surface area is a measurement of the total area of the outer surface of an object or shape. It is the sum of the areas of all the individual surfaces that make up the object.
The concept of surface area applies to both two-dimensional shapes (such as polygons) and three-dimensional objects (such as cubes, spheres, cylinders, and prisms).
To determine the surface area of rotating the curve [tex]x=2\sqrt{a^2-x^2}[/tex]around the y-axis, we can use the formula for the surface area of revolution.
The formula for the surface area of revolution when rotating a curve y=f(x) around the x-axis over an interval [a,b] is given by:
[tex]S=2\pi \int\limits^b_a f(x)\sqrt{ 1+(\frac{dy}{dx})^2} dx[/tex]
In this case, the given curve is[tex]x=2\sqrt{a^2-x^2}[/tex] , and we need to rotate it around the y-axis over the interval [tex]0\leq y\leq \frac{a}{2}[/tex].
First, let's find the derivative [tex]\frac{dy}{dx}[/tex] using implicit differentiation. Differentiating[tex]x=2\sqrt{a^2-x^2}[/tex] with respect to y, we get:
[tex]\frac{dy}{dx} =\frac{-2y}{\sqrt{a^2-x^2} }[/tex]
Next, we substitute the values into the surface area formula:
[tex]S=2\pi \int\limits^\frac{a}{2} _0 2\sqrt{a^2-x^2} \sqrt{ 1-(\frac{-2y}{\sqrt{a^2-y^2}})^2} dy[/tex]
Simplifying the expression inside the square root:
[tex]S=2\pi \int\limits^\frac{a}{2} _0 2\sqrt{a^2-y^2} \sqrt{ 1+\frac{4y^2}{{a^2-y^2}}} dy[/tex]
Combining the terms inside the square root:
[tex]S=2\pi \int\limits^\frac{a}{2} _0 2\sqrt{a^2-y^2} \sqrt{ \frac{a^2}{{a^2-y^2}}} dy\\[/tex]
Simplifying further:
[tex]S=2\pi \int\limits^\frac{a}{2} _0 2a dy[/tex]
Evaluating the integral:
[tex]S=2\pi [2ay]^\frac{a}{2}_0[/tex]
[tex]S=2\pi [2a.\frac{a}{2}-2a.0]\\S=2\pi .a^2[/tex]
Therefore, the surface area of rotating the curve [tex]x=2\sqrt{a^2-x^2}[/tex] over the y-axis over the interval [tex]0\leq y\leq \frac{a}{2}[/tex] is [tex]2\pi a^{2}[/tex].
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