As with any problem like this, it is fairly easy to make a little mistake and not get the exact final answer. For that reason, I encourage you to go through my steps to check for any minor errors.
This is a work problem with integrals. Each horizontal slice is a circle with area = pi*r^2. The radius of the slice depends on the y-coordinate. We can solve for the radius in terms of y using the ratio 4/6 = r/y. So r = 2y/3. Then we integrate from 0 to 5 the integrand: 62.4*pi*(2y/3)^2. The antiderivative is 62.4*pi*4/27*y^3. Plugging in 5 and 0, we get 3630.28 ft-lb. And plugging in 6 and 2, we get 6040.79 ft-lb.
Please let me know if you need any clarification. I'm always happy to answer your questions.