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November 10th, 2014, 11:02 AM   #1
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volume x-axis vs y-axis

Hello again Math Forum community! I am already liking this bunch of people.

So I am working on some volume problems and I am understanding most of context, but I am getting hung up on which formula to use for x vs y axis rotations.

Example:

"Find the volume of the region by revolving the region about the y-axis". The region enclosed by $\displaystyle x=y^\frac13, x=0, y=8$
A. $\displaystyle \frac{96}5\pi$

B. $\displaystyle 32\pi $

C. $\displaystyle 16\pi $

D. $\displaystyle 12\pi$

So which values do I use for the bounds of the integral? $\displaystyle \int_0^{y^\frac13}$?????

And will the formula look like the following?:

$\displaystyle \int_0^{y^\frac13}2\pi x {8}$


Do I have this backwards?
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November 10th, 2014, 12:21 PM   #2
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Correct answer is A.
Use the formula:

Last edited by skaa; November 10th, 2014 at 12:23 PM.
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November 10th, 2014, 12:23 PM   #3
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$\displaystyle V=\pi \int_{0}^{8}(y^{\frac{1}{3}})^2dy=\pi \int_{0}^{8}y^{\frac{2}{3}}dy=\frac{3}{5}\pi y^{\frac{5}{3}} |_0 ^8$
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