We need to find where these two curves intersect to find the bounds of integration.
y^2=x^3 and y=x^2, squaring the second expression, y^2=x^4 Solving for y^2,.......... [x^4=x^3] i.e, x^3[x-1]=0.
So x=1, x=0 are the points of intersection.
From the graphs of these expressions in can be seen that y=sqrt[x^3] has a greater area than y= x^2 so we must find the area under y =x^2 and subtract it from the area under y=sqrt[x^3] and then revolve this area about the x axis between the bounds x=1, x= 0
Volume of revolution is given by piint_a^by^2dx So the volume of revolution = [piint_0^1x^3dx - piint_0^1x^4dx].= pi[x^4/4- x^5/5] [ after integration by the general power rule] and evaluated for x=1, x=0 will result in pi[ 1/4-1/5] = pi/20.
