Expert: Abe Mantell - 8/6/2006

Question
Hi,

I'm trying to find the ways to get these volumes using the washer and shell method.  The region is bound by y=x^3, and the x-axis with 0<=x<=2.  Could you help me?  

For the washer method revolved around the y axis i got the answer 96pi over 5.  I cant seem to set up the shell method to get that.  And maybe that answer is incorrect.  

Also, I'm trying to find the volume by revolving the same region about y=8.  I'm getting a weird answer for that... 320pi, which i dont think is correct.  Any help would be great.  Thanks,

J

Answer
Hello,

Washer Vethod:
V=Integral(pi*(2^2-[y^(1/3)]^2),y=0..2^3)
=Integral(pi*(4-y^(2/3)),y=0..8)=64pi/5

Shell Method:
V=Integral(2*pi*x*x^3, x=0..2)=64pi/5

For rotating about y=8 (shell method):
V=Integral(pi*(8^2-(8-x^3)^2),x=0..2)
=Integral(16x^3-x^6,x=0..2)=320pi/7

OK?

Abe