Volume

Jul 2008
15
5
Consider the given curves to do the following. 8 y = x^3, y = 0 , x = 4 Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 8.
 
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skeeter

MHF Helper
Jun 2008
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North Texas
Consider the given curves to do the following. 8 y = x^3, y = 0 , x = 4 Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 8.
did you sketch a graph ?

\(\displaystyle
V = 2\pi \int_0^8 (8-y)(4 - 2\sqrt[3]{y}) \, dy
\)


check the result by using washers ...

\(\displaystyle
V = \pi \int_0^4 8^2 - \left(8 - \frac{x^3}{8}\right)^2 \, dx
\)
 
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Jul 2008
15
5
I keep getting (1152 * Pi)/7. I am confused about why you have 4 - 2y^(1/3) when using the method of cylindrical shells. For the washer method
it makes sense. The outer radius is 8 and the inner is 8 - x^3/8.

I tried checking it with Maple and it still spits out the answer the answer from above. Maple just integrates Pi*(8-x^3/8)^2 over the interval x = 0 to x = 4.

Thank you for your time
 
Jul 2008
15
5
I see

I see where my error is. It is bounded by x = 4.
Thank you very much.