It usually helps to draw these type problems and shade the section that will be rotating. Rewrite the y^3 = x as y = x^(1/3). These two curves intersect at x = 0 and x = 1.17, which can be found graphically or by setting the two equations equal to each other. The section enclosed by the curves above the y axis is between x= 0 and x = 1.17.
To calculate the volume, you take the integral of the area. If you draw the 3D version of this, you'll see that we'll be looking for the volume of the walls of a cone-type shape. The cross-section of this shape will be a ring. The outer radius is y = x^(1/3) and the inner radius is y = 0.9x. The area is then π((outer radius)^2-(inner radius)^2). Area = π((x^(1/3))^2 - (0.9x)^2 = x^(2/3) - 0.81x^2.
Then, take the integral of this area from x = 0 to x = 1.17. π is a constant that can be pulled out of the integral. Use the power rule for the other two components of the integral.
The indefinite integral is V = π*(0.6x^(5/3) - 0.27x^3), then solve within the bounds from 0 to 1.17. The volume is 1.09.
