# Volume by Cylindrical shells

## Homework Statement

Rotate around the y-axis the region above the graph of y=x3 that is bounded by the lines x=1 and y=8

## Homework Equations

dV= (2pix)(y)(dx)

## The Attempt at a Solution

dV = (2pix)(y)dx
dV = (2pix)(x^3) dx
= 2pix^4

I integrated from y = 1 to y=8 and I get the wrong answer.

V = 2∏x=1x=2x^4 dx
V = 2∏[1/5x^5]21

Can someone please explain to me, how I would set up this equation? Something to do with the first function subtracting the second function which I think is x=1

Here is a graph: http://www3.wolframalpha.com/Calculate/MSP/MSP15491a07c8557bi4590b00001ibd52eb6gcc3177?MSPStoreType=image/gif&s=21&w=366&h=298&cdf=RangeControl [Broken]

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SammyS
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Homework Helper
Gold Member

## Homework Statement

Rotate around the y-axis the region above the graph of y=x3 that is bounded by the lines x=1 and y=8

## Homework Equations

dV= (2pix)(y)(dx)

## The Attempt at a Solution

dV = (2pix)(y)dx
dV = (2pix)(x^3) dx
= 2pix^4

I integrated from y = 1 to y=8 and I get the wrong answer.

V = 2∏x=1x=2x^4 dx
V = 2∏[1/5x^5]21

Can someone please explain to me, how I would set up this equation? Something to do with the first function subtracting the second function which I think is x=1

Here is a graph: http://www3.wolframalpha.com/Calculate/MSP/MSP15491a07c8557bi4590b00001ibd52eb6gcc3177?MSPStoreType=image/gif&s=21&w=366&h=298&cdf=RangeControl [Broken]
The problem is with the following:
dV= (2πx)(y)(dx)
The height of the cylindrical shell is 8-y, not y.

So dV = (2πx)(8-y)(dx)

Last edited by a moderator:
The problem is with the following:
dV= (2πx)(y)(dx)
The height of the cylindrical shell is 8-y, not y.

So dV = (2πx)(8-y)(dx)

[STRIKE]So dV = (2∏x)(8-x^3)dx
= 16∏x-2∏x^4 dx

V= 2∏∫(8x-x^4) dx ?

[/STRIKE]

Thank you for helping me!