The region bounded by the parabola y=4x-x^2 and the x-axis is revolved about the x-axis. Find the volume of the resulting solid.

Question

Calculus- Volume of solid

Answer 1

The curve intersects the x-axis at (0,0) and (0,4). Consider the volume as a stack of thin discs of thickness dx. The volume of a thin disc is πr² so, adding up the volumes of all those thin discs, we get

v = ∫[0,4] πy² dx = ∫[0,4] π(4x-x²)² dx = 512π/15

Or, you could use nested shells of thickness dy. Making x a function of y is a bit harder, since it has two branches. But the symmetry of the region means we can just double the volume we get by rotating just the left half. The volume of a thin shell of thickness dy is 2πrh, where h=2-x, so if we add up all the cylinders, we get

v = 2∫[0,4] 2πy√(4-y) dy = 512π/15