Law Curve Nozzle

The nozzle shown below is composed of the region bounded by these two functions:

Outer Radius  R(x) = -x+3

Inner Radius r(x) = x^2 + 1

For x values from -2 to 1

The region is revolved 360 degrees about the X axis (the illustration below is cut away for clarity).

Model the nozzle using two law curves.

Verify that the volume of this shape is equal to the definite integral


Which evaluates to



Adapted from:

Thomas, George B., and Ross L. Finney. Calculus and Analytic Geometry. Reading, MA: Addison-Wesley, 1992. 333. Print.