Finding the shell height in the Shell Method

• Sep 26th 2009, 01:28 PM
mathgeek777
Finding the shell height in the Shell Method
Good afternoon.

I am working on a problem very similar to the following.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Find the volume of the solid generated by revolving $\displaystyle x=3-y^2$, bounded by the x-axis, the y-axis, x=3 , and $\displaystyle y=\sqrt{2}$, about the x-axis.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Based on the information given in this problem, I have been able to set up the integral to show what is below.

$\displaystyle \int_0^3 2\pi r (3-y^2) dy$

My one problem is the r in the integral. I know how to obtain it when the region is being revolved around the y-axis, but I am not sure how to obtain it when the region is being revolved around the x-axis. Is it the distance from the x-axis to the top of the bounded region, or do you have to obtain it via another method?

Thank you in advance for any assistance given.
• Sep 26th 2009, 01:40 PM
skeeter
Quote:

Originally Posted by mathgeek777
Good afternoon.

I am working on a problem very similar to the following.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Find the volume of the solid generated by revolving $\displaystyle x=3-y^2$, bounded by the x-axis, the y-axis, x=3 , and $\displaystyle y=\sqrt{2}$, about the x-axis.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Based on the information given in this problem, I have been able to set up the integral to show what is below.

$\displaystyle \int_0^3 2\pi r (3-y^2) dy$

My one problem is the r in the integral. I know how to obtain it when the region is being revolved around the y-axis, but I am not sure how to obtain it when the region is being revolved around the x-axis. Is it the distance from the x-axis to the top of the bounded region, or do you have to obtain it via another method?

Thank you in advance for any assistance given.

$\displaystyle r$ = distance from the x-axis to a representative shell ...
$\displaystyle r = y$ in this case.

also, check your limits of integration ...

$\displaystyle V = 2\pi \int_0^{\sqrt{2}} y(3-y^2) \, dy$

note the attached graph.
• Sep 26th 2009, 01:47 PM
mathgeek777
Actually, I misspoke when I stated the upper bound of y. It was actually $\displaystyle y=\sqrt{3}$, but it is irrelevant for the purposes of this question. Thanks for the help skeeter.