Question

Find the volume of the solid of revolution by roating the region R about the y-axis. Write the answer as a multiple of pi.
y= sqrt(x), y=x^2, in the first Quadrant

Answer 1

The curves intersect at (0,0) and (1,1).

Using shells,

v = ∫2πrh dx [0,1]
where r = x and h=√x-x^2
v = 2π∫x(√x-x^2) dx [0,1]
= 3π/10

Using discs,

v = ∫π(R^2-r^2) dy [0,1]
where R=√y and r=y^2
v = π∫(y-y^4) dy [0,1]
= 3π/10