Question

1) A region is bounded by the line y = x and the parabola y = x2 - 6x + 10. What is the volume of the solid generated by revolving the region about the x-axis?

Answer 1

First define the x limits of the enclosed region. They are between the two roots of
x^2 -6x +10 = x
x^2 -7x +10 = 0
(x - 5)(x -2)= 0

You want to integrate between x = 2 and 5.

The function that you integrate is
f(x) = pi*{-[x^2 -6x +10]^2 + x^2} dx.
The y = x curve lies above the parabola in the interval