Question

Let A be the region in the plane consisting of those points (x,y) such that 0≤y≤1−x^2. Rotate A around the x-axis in space. What is the volume of the resulting solid?

Answer 1

using discs,

v = ∫[-1,1] πr^2 dx
where r = y = 1-x^2
v = π∫[-1,1] (1-x^2)^2 dx
...
= 16π/15

using shells,
v = 2∫[0,1] 2πrh dy
where r=y and h=x=√(1-y)
v = 4π∫[0,1] y√(1-y) dy
...
= 16π/15