Cracking The Ap Calculus ab Exam 2018

From a general standpoint: If we have a region whose area is bounded above by the curve y = f(x) and
below by the curve y = g(x), on the interval [a, b], then each cylinder will have a height of f(x) − g(x), a
radius of x, and an area of 2πx[f(x) − g(x)].

To find the volume, evaluate the integral.

2 π x[(f(x) −   g(x)]   dx

This is the formula for finding the volume using cylindrical shells when the region is rotated around the y-
axis.

Example 5: Find the volume of the region that results when the region bounded by the curve y = , the

x-axis, and the line x = 9 is revolved about the y-axis. Set up but do not evaluate the integral.

Your sketch should look like the following:

Notice that the limits of integration are from x = 0 to x = 9, and that each vertical slice is bounded from

above by the curve y = and from below by the x-axis (y = 0). We need to evaluate the integral.

2 π x(  −   0)  dx  =   2π x( ) dx

Example 6: Find the volume that results when the region in Example 5 is revolved about the line x = −1.
Set up but do not evaluate the integral.