Find the volume generated by rotating about the given line

Question

a) y^2=x, x=2y; about the y-axis

b) y=x, y=sqrtx; about y=1

Answer 1

(a)
using shells of thickness dx,
v = ∫[0,4] 2πrh dx
where r=x and h=√x-x/2
v = ∫[0,4] 2πx(√x-x/2) dx = 64π/15

using discs of thickness dy,
v = ∫[0,2] π(R^2-r^2) dy
where R=2y and r=y^2
v = ∫[0,2] π((2y)^2-y^4) dy = 64π/15

Now you try (b), recalling the formulas for volumes of discs and shells, above. Sketch the curves and figure out the radius or height.