Find the volume for the region bounded by y = 0, y = 1, x = 1, and y = x - 2, by revolving R about the y axis?

Answer 1

Rotate every equation to make a shell method problem a disk method problem.

Replace y with -x, and x with y.

-x = 0 => x = 0

-x = 1 => x = -1

y = 1 = r

-x = y - 2 => y = 2 - x = R

Now, we can rotate about the x - axis.

For volumes of revolution about the x - axis, the cross-sections of the solid are all circles.

Using Cavalieri’s principle,

The volume for the region bounded by y = 0, y = 1, x = 1, and y = x - 2, by revolving R about the y axis is 16π/3 cubic units.