Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis. y
Answers
Answer:
[tex]Volume = \frac{384}{7}\pi[/tex]
Step-by-step explanation:
Given (Missing Information):
[tex]y = x^\frac{3}{2}[/tex]; [tex]y = 8[/tex]; [tex]x=0[/tex]
Required
Determine the volume
Using Shell Method:
[tex]V = 2\pi \int\limits^a_b {p(y)h(y)} \, dy[/tex]
First solve for a and b.
[tex]y = x^\frac{3}{2}[/tex] and [tex]y = 8[/tex]
Substitute 8 for y
[tex]8 = x^\frac{3}{2}[/tex]
Take 2/3 root of both sides
[tex]8^\frac{2}{3} = x^{\frac{3}{2}*\frac{2}{3}}[/tex]
[tex]8^\frac{2}{3} = x[/tex]
[tex]2^{3*\frac{2}{3}} = x[/tex]
[tex]2^2 = x[/tex]
[tex]4 =x[/tex]
[tex]x = 4[/tex]
This implies that:
[tex]a = 4[/tex]
For [tex]x=0[/tex]
This implies that:
[tex]b=0[/tex]
So, we have:
[tex]V = 2\pi \int\limits^a_b {p(y)h(y)} \, dy[/tex]
[tex]V = 2\pi \int\limits^4_0 {p(y)h(y)} \, dy[/tex]
The volume of the solid becomes:
[tex]V = 2\pi \int\limits^4_0 {x(8 - x^{\frac{3}{2}}}) \, dx[/tex]
Open bracket
[tex]V = 2\pi \int\limits^4_0 {8x - x.x^{\frac{3}{2}}} \, dx[/tex]
[tex]V = 2\pi \int\limits^4_0 {8x - x^{\frac{2+3}{2}}} \, dx[/tex]
[tex]V = 2\pi \int\limits^4_0 {8x - x^{\frac{5}{2}}} \, dx[/tex]
Integrate
[tex]V = 2\pi * [{\frac{8x^2}{2} - \frac{x^{1+\frac{5}{2}}}{1+\frac{5}{2}}]\vert^4_0[/tex]
[tex]V = 2\pi * [{4x^2 - \frac{x^{\frac{2+5}{2}}}{\frac{2+5}{2}}]\vert^4_0[/tex]
[tex]V = 2\pi * [{4x^2 - \frac{x^{\frac{7}{2}}}{\frac{7}{2}}]\vert^4_0[/tex]
[tex]V = 2\pi * [{4x^2 - \frac{2}{7}x^{\frac{7}{2}}]\vert^4_0[/tex]
Substitute 4 and 0 for x
[tex]V = 2\pi * ([{4*4^2 - \frac{2}{7}*4^{\frac{7}{2}}] - [{4*0^2 - \frac{2}{7}*0^{\frac{7}{2}}])[/tex]
[tex]V = 2\pi * ([{4*4^2 - \frac{2}{7}*4^{\frac{7}{2}}] - [0])[/tex]
[tex]V = 2\pi * [{4*4^2 - \frac{2}{7}*4^{\frac{7}{2}}][/tex]
[tex]V = 2\pi * [{64 - \frac{2}{7}*2^2^{*\frac{7}{2}}][/tex]
[tex]V = 2\pi * [{64 - \frac{2}{7}*2^7][/tex]
[tex]V = 2\pi * [{64 - \frac{2}{7}*128][/tex]
[tex]V = 2\pi * [{64 - \frac{2*128}{7}][/tex]
[tex]V = 2\pi * [{64 - \frac{256}{7}][/tex]
Take LCM
[tex]V = 2\pi * [\frac{64*7-256}{7}][/tex]
[tex]V = 2\pi * [\frac{448-256}{7}][/tex]
[tex]V = 2\pi * [\frac{192}{7}][/tex]
[tex]V = [\frac{2\pi * 192}{7}][/tex]
[tex]V = \frac{\pi * 384}{7}[/tex]
[tex]V = \frac{384}{7}\pi[/tex]
Hence, the required volume is:
[tex]Volume = \frac{384}{7}\pi[/tex]
The volume of the solid of revolution generated by revolving the plane region about the y-axis is [tex]\frac{10240\pi}{3}[/tex] cubic units.
How to calculate the volume of a solid of revolution by shell method
Let be [tex]f(x) = x^{5/2}[/tex] and [tex]g(x) = 32[/tex], whose point of intersection is [tex](x,y) = (4, 32)[/tex]. Solid of revolution generated by revolving the plane region about the y-axis is defined by the following formula:
[tex]V = 2\pi\int\limits^{32}_0 {y\cdot g(y)} \, dy[/tex] (1)
If we know that [tex]g(y) = y^{2/5}[/tex], then the volume of the solid of revolution is:
[tex]V = 2\pi\int\limits^{32}_{0} {y^{7/5}} \, dy[/tex]
[tex]V = 2\pi\cdot \left(\frac{5}{12}\cdot y^{12/5}\right)\left|_{0}^{32}[/tex]
[tex]V = \frac{5\pi}{6} \cdot y^{12/5}|\limit_{0}^{32}[/tex]
[tex]V = \frac{5\pi}{6}\cdot 32^{12/5}[/tex]
[tex]V = \frac{10240\pi}{3}[/tex]
The volume of the solid of revolution generated by revolving the plane region about the y-axis is [tex]\frac{10240\pi}{3}[/tex] cubic units. [tex]\blacksquare[/tex]
Remark
The statement is incomplete. Complete form is shown below:
Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis. [tex]y = x^{5/2}[/tex], [tex]y = 32[/tex], [tex]x = 0[/tex].
To learn more on solids of revolution, we kindly invite to check this verified question: https://brainly.com/question/338504
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Answer:
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Step-by-step explanation:
Recall that fractional exponents are other form of representing roots.
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Answer:
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Step-by-step explanation:
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Answers
Answer: 75.333333...
Step-by-step explanation: Might want to round that number to the nearest tenths. This number is a repeating digit.
Here we will show you step-by-step with detailed explanation how to calculate 452 divided by 6 using long division.
Before you continue, note that in the problem 452 divided by 6, the numbers are defined as follows:
452 = dividend
6 = divisor
Step 1:
Start by setting it up with the divisor 6 on the left side and the dividend 452 on the right side like this:
6 ⟌ 4 5 2
Step 2:
The divisor (6) goes into the first digit of the dividend (4), 0 time(s). Therefore, put 0 on top:
0
6 ⟌ 4 5 2
Step 3:
Multiply the divisor by the result in the previous step (6 x 0 = 0) and write that answer below the dividend.
0
6 ⟌ 4 5 2
0
Step 4:
Subtract the result in the previous step from the first digit of the dividend (4 - 0 = 4) and write the answer below.
0
6 ⟌ 4 5 2
- 0
4
Step 5:
Move down the 2nd digit of the dividend (5) like this:
0
6 ⟌ 4 5 2
- 0
4 5
Step 6:
The divisor (6) goes into the bottom number (45), 7 time(s). Therefore, put 7 on top:
0 7
6 ⟌ 4 5 2
- 0
4 5
Step 7:
Multiply the divisor by the result in the previous step (6 x 7 = 42) and write that answer at the bottom:
0 7
6 ⟌ 4 5 2
- 0
4 5
4 2
Step 8:
Subtract the result in the previous step from the number written above it. (45 - 42 = 3) and write the answer at the bottom.
0 7
6 ⟌ 4 5 2
- 0
4 5
- 4 2
3
Step 9:
Move down the last digit of the dividend (2) like this:
0 7
6 ⟌ 4 5 2
- 0
4 5
- 4 2
3 2
Step 10:
The divisor (6) goes into the bottom number (32), 5 time(s). Therefore put 5 on top:
0 7 5
6 ⟌ 4 5 2
- 0
4 5
- 4 2
3 2
Step 11:
Multiply the divisor by the result in the previous step (6 x 5 = 30) and write the answer at the bottom:
0 7 5
6 ⟌ 4 5 2
- 0
4 5
- 4 2
3 2
3 0
Step 12:
Subtract the result in the previous step from the number written above it. (32 - 30 = 2) and write the answer at the bottom.
0 7 5
6 ⟌ 4 5 2
- 0
4 5
- 4 2
3 2
- 3 0
2
You are done, because there are no more digits to move down from the dividend.
The answer is the top number and the remainder is the bottom number.
Therefore, the answer to 452 divided by 6 calculated using Long Division is:
75
2 Remainder
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Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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Answers
9514 1404 393
Answer:
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Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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Answer:
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Step-by-step explanation:
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Equation/Question:
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Answer/Way I did it:
1. Remove parentheses.
-6x+8=2
2. Subtract 8 from both sides.
-6x=2-8
3. Simplify 2-8 to -6.
-6x=-6
4. Divide both sides by -6.
x=1
-------------
Checking:
Equation/Question:
−6x−(−8)=2
----------------------
1. Let x=1.
−6×1+8=2
2. Simplify -6×1 to −6.
-6+8=2
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Answers
9514 1404 393
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Step-by-step explanation:
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x = -12
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The explicit rule for the geometric sequence is:
Answers
Answer:
nth term = 1000×1.2ⁿ⁻¹
Step-by-step explanation:
The explicit rule for the given geometric sequence is aₙ = a₁rⁿ⁻¹ = 1000 x 1.2ⁿ⁻¹
What is a geometric sequence?A geometric sequence is a special type of sequence where the ratio of every two successive terms is a constant.
Given that, in two years the deer population in a state forest was 1,200. In year 4, the population was 1,728,
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Answer · 1 vote
Formula, a^3 - b^3 = (a - b)(a^2 + ab + b^2) a^3 + b^3