Write the integral in one variable to find the volume of the solid obtained by rotating the first‐quadrant region bounded by y = 0.5x2 and y = x about the line x = 5.

Answer 1

The curves intersect at (0,0) and (2,2)

So, using shells of thickness dx,
v = ∫[0,2] 2πrh dx
where r=5-x and h=x-x^2/2
v = ∫[0,2] 2π(5-x)(x-x^2/2) dx = 16π/3

using discs (washers) of thickness dy,
v = ∫[0,2] π(R^2-r^2) dy
where R=5-y and r=5-√(2y)
v = ∫[0,2] π((5-y)^2-(5-√(2y))^2) dy = 16π/3

Answer 2

Thank you!