Check your limits for the shells
I have the function f(x)=x^{2}
I am to give the volume of the shape when this function is spun 360 degrees around the x axis by using the disc method, and the shell method, from x=1 and x=3.
When I work out the disc method, I get (242pi)/5. But when I use the shell method, I get (232pi)/5. I have been looking over my work, and it all seems to work out, so I am curious why they are different. I will show my setups below:
Disc method:
v=pi*r^{2}*h
r=y=f(x)=x^{2}
h=dx
v=pi(x^{4}) dx
Then I integrate and get that v=pi[(1/5)x^{5}]^{3}_{1 }which gives me (242pi)/5.
Shell method:
v=2pi*r*h dy
r=y
h=3-f(y)=3-sqrt(y)
v=2pi(3y-y^{3/2}) dy
I then integrate, and get that v=2pi[(3/2)y^{2}-(2/5)y^{5/2}]^{9}_{1} which then gives me (232pi)/5.
Have I done something wrong here? Or is there a reason for the discrepancy?
As tom said, your limits of integration are wrong for the shell method. Additionally, the h you use is only correct on the limits you currently have for the integration.
Hint: Currently, you are finding the volume of a shape that has a cylindrical hole through the center.