Let r r be the region enclosed by x=2 x = 2 , x=3 x = 3 , y=16-x^4 y = 16 − x 4 and y=0 y = 0 . find the volume of the solid obtained by rotating r r about y y -axis.

You have to use the "Cylinder Method" (aka "Shell Integration").
The volume of the solid obtained by rotating the region between f(x), y=0, x=a, and x=b is given by:



In your case, . So, substituiting f(x), a and b, and taking into account that in the interval (2,3) we can assume , and, therefore, , we get:



We have made use of the fact that you can take constant factors out of the integral.

Then, being the integral of the sum equal to the sum of the integrals:



For both integrals we'll use the power integration rule, and the Second Fundamental Theorem of Calculus: