Check my work: write the integral in one variable to find the volume of the solid obtained by rotating the first‐quadrant region bounded by y = 0.5x^2 and y = x about the line x = 5. v = ∫[0,2] 2πrh dx r=5-x and h=x-x^2/2 v = ∫[0,2] 2π(5-x)(x-x^2/2) dx = 16π/3 v = ∫[0,2] π(r^2-r^2) dy r=5-y and r=5-√(2y) v = ∫[0,2] π((5-y)^2-(5-√(2y))^2) dy = 16π/3