Determining volumes by slicing

Question

Find the exact value of the volume of the object we obtain when rotating  about the -axis, where  is the finite region bounded by  and .

Answer 1

You can apply washer method.

In washer method rotating about a horizontal line, you should integrate with respect to x.

To find your limits, the intersection points, set the curves equal each other. 4x+7=x²-4x+22. Solving the equation,you get x=3 and x=5.

Note that the function f(x) is on the top of the function g(x). Hence,

the volume = ∫⁵₃ [ (4x+7)² - (x² -4x + 22)² ] dx