Volume of solid by rotating it

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = x2, x = y2; about y = 1

Answer 1

I assume that the curves are y=x² and x=y². Intersections of the curves f(x)=x² and g(x)=√x are points (0, 0) and (1, 1). When revolving around line y=1, the volume of revolution is calculated as

 

V=π∫₀¹[(f(x)-1)²-(g(x)-1)²]dx=

=π∫₀¹[(x²-1)²-(√x-1)²]dx=π∫₀¹(x⁴-2x²-x+2√x)dx=

=π(1/5-2/3-1/2+4/3)=11/30