Question

Set up the simplified integral and compute the volume created when the area bounded by y=x^2-1 and y=3 is rotated around the y-axis.

Answer 1

think of the volume as a lot of thin cylinders of thickness dx.

v = ∫[0,2] 2πrh dx
where r = x and h = 3-y = 4-x^2
v = ∫[0,2] 2πx(4-x^2) dx

or, you can think of the volume as a stack of thin discs of thickness dy. Then

v = ∫[-1,3] πr^2 dy
where r = x = √(y+1)
v = ∫[-1,3] π(y+1) dy