Question

How do you find the volume of the solid generated by revolving the region enclosed by the parabola `y^2=4x` and the line y=x revolved about the x-axis?

Answer 1

The problem is equivalent to:

find the volume of the solid generated by revolving the region enclosed by the parabola `y = x^2/4` and the line `x = y` both revolted about the `y` axis. So
`f_1(r)=r^2/4`
`f_2(r)=r`
`V_1(R) = 2pi int_0^R f_1(r)rdr`
`V_2(R) = 2pi int_0^R f_2(r)rdr`
`V(R) = V_2(R)-V_1(R)`
`R` is such that `f_1(r) = f_2(r)` giving `R = 4`
So `V(4) = 32pi/3`