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 = 7. (10 points)

Answer 1

The curves intersect at (0,0) and (2,2)
using shells of thickness dx,
v = ∫[0,2] 2πrh dx
where r = 7-x and h=x - 0.5x^2
v = ∫[0,2] 2π(7-x)(x - 0.5x^2) dx = 8π

using discs (washers) of thickness dy,
v = ∫[0,2] π(R^2-r^2) dy
where R = 7-y and r = 7-√(2y)
v = ∫[0,2] π((7-y)^2-(7-√(2y))^2) dy = 8π

Answer 2

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