How to Find Volume by Cylindrical Shells?

[Biosyn]

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=1∫^x=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

 

[SammyS]

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=1∫^x=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

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)

 

[Biosyn]

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!