AP Calculus AB Exam Question:

A solid is obtained by revolving the region bounded by the curve given by the equation y = x^2 + 1, the x-axis, and the vertical lines x = 0 and x = 2 about the x-axis. Determine the volume of the solid.

Solution:

To find the volume of the solid, we will make use of the method of cylindrical shells.

Step 1:

First, let's graph the given function y = x^2 + 1 and identify the region bounded by the curve, the x-axis, and the vertical lines x = 0 and x = 2.

The region we are interested in is shaded in the graph.

Step 2:

To apply the method of cylindrical shells, we need to express the volume as an integral in terms of the height of the shells.

Consider a thin vertical strip of width dx at a distance x from the y-axis. The volume of the corresponding cylindrical shell is approximately given by:

where f(x) is the upper boundary curve (y = x^2 + 1) and g(x) is the lower boundary (x-axis).

Step 3:

We need to determine the limits of integration for the variable x. Since the region is bounded by x = 0 and x = 2, our integral will be evaluated from x = 0 to x = 2.

Step 4:

Now, let's write the integral expression for the total volume V of the solid:

Simplifying the integrand:

Step 5:

Integrate the expression with respect to x:

Calculating the definite integral:

Simplifying:

Step 6:

Therefore, the volume of the solid is 20π cubic units.