Finding the shell height in the Shell Method

**mathgeek777** · Sep 26th 2009, 01:28 PM

Good afternoon.

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

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

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

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

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.

**skeeter** · Sep 26th 2009, 01:40 PM

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 $x=3-y^2$ , bounded by the x-axis, the y-axis, x=3 , and $y=\sqrt{2}$ , 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.

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

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.

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

also, check your limits of integration ...

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

note the attached graph.

**mathgeek777** · Sep 26th 2009, 01:47 PM

Actually, I misspoke when I stated the upper bound of y. It was actually $y=\sqrt{3}$ , but it is irrelevant for the purposes of this question. Thanks for the help skeeter.

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