Question

Find the volume of the solid obtained by rotating the region bounded by the curves y=x^6, y=1, about the line y=6

Answer 1

This is the situation: (see image)

We can simplify a bit by moving everything 6 units down and rotate about the x-axis.
If we use the cylindrical shell method, we get the integral\[V= \int\limits_{-1}^{1}\pi(f(x) - 6)^2dx=2\pi \int\limits_{0}^{1}(x^6-6)^2dx\]I saved on the calculations (and errors!) by using the symmetry of the thing: the integral from -1 to 1 is equal to twice the integral from 0 to 1.
Because \(\pi\) is a constant, it can also be put in front of the integral.

Now all you have to do is expand the brackets and integrate every term.