Answer:
2pi/15
Step-by-step explanation:
The intersection of y=x and y=x^2 is (0,0) and (1,1)
Washer method
Prefered this method for this one since slices are perpendicular to axis of rotation
Integrate(Big circle - Small Circle)dx
Integrate(pi*(x)^2-pi*(x^2)^2)dx on x=0..1
Integrate(pi*x^2-pi*x^4)dx on x=0..1 is 2pi/15
Shells method
So I have to solve for x since I will be integrating with respect to y
The height of the the shell will be sqrt(y)-y
The radius will be y since that is the distance between axis of rotation in a point within the shell
Integrate(2*pi*r*h ) dy
Integrate(2*pi*(sqrt(y)-y)*y) dy on y=0..1 is 2pi/15
