can anyone show me how to do this calculus problem?

Question

It is: find the volume using shell method for the graph y=x^3, bound by x=1, x=2, and y=0, and revolved around the x-axis. thanks in advance!

Answer 1

Shell method is a real pain for this question...

Notice we have one big cylinder in the middle, radius 1, height 1, so volume is π.
Then, for y from 1 to 8, if we have a cylindrical shell of width dy and radius y, the height is 2 - y^(1/3), so the volume is &2pi;y(2 - y^(1/3)) dy.
So the total volume is
π + π∫(1 to 8) 4y - 2y^(4/3) dy
= π (1 + [2y^2 - (6/7)y^(7/3)][1 to 8])
= π (1 + 8^2(2 - (6/7).8^(1/3)) - 1^2(2 - (6/7).1^(1/3)))
= π (1 + 64(2/7) - 1(8/7))
= 127π/7.

Compare the disk method:
V = π∫(1 to 2) y^2 dx
= π∫(1 to 2) x^6 dx
= π (2^7 - 1)/7
= 127π/7.
Much easier in this case, isn't it?