Thread: 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.

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.

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.