Question

Find the volume of the solid of revolution obtained by revolving region bounded by the parabolas 2y=x^2 and y^2=4x about the x-axis

Answer 1

The curves intersect at (0,0) and (2∛2,2∛4)

using discs,
v = ∫[0,2∛2] π(R^2-r^2) dx
where R=√(4x) and r=x^2/2
v = ∫[0,2∛2] π(4x-x^4/4) dx = 24π∛4/5

using shells,
v = ∫[0,2∛4] 2πrh dy
where r=y h=(y^2/4)-√(2y)
v = ∫[0,2∛4] 2πy(√(2y)-(y^2/4)) dy = 24π∛4/5