Rotated about x-axis, shell method, y = 8, x = 0
formula: 2pixy
xy = x(x^3) = x^4
integrate:
x^4 = 1/5 x^5
1/5(8)^5 = 6553.6
*2 = 13107.2 pi
answer should be 768/7 pi though.
I assume you mean that y= x^3 is being rotated about the x-axis. it would have been nice to say that!
However, the formula is NOT "2pi xy". That would be correct is you were rotating the region above y= x^3, but for the region below, using the "shell method", each shell would be a cylinder with radius y and height 2- x, integrated with respect to y, not x. You want to integerate 2piy(2- x)dy.
No, x^4 is NOT equal to (1/5)x^5! Yes, of course, you mean that the integral of x^4 is equal to that (well, equal to (1/5)x^5+ C), but say that- don't write non-sense and depend on others to guess what you mean.integrate:
x^4 = 1/5 x^5
1/5(8)^5 = 6553.6
*2 = 13107.2 pi
answer should be 768/7 pi though.