Cracking The Ap Calculus ab Exam 2018

Suppose the region we’re interested in is revolved around the y-axis instead of the x-axis. Now, to find
the volume, you have to slice the region horizontally instead of vertically. We discussed how to do this in
the previous unit on area.

Now, if you have a region whose area is bounded on the right by the curve x = f(y) and on the left by the
curve x = g(y), on the interval [c, d], then each washer has an area of

π[f(y)^2    −   g(y)^2 ]

To find the volume, evaluate the integral.

π [f(y)^2   −   g(y)^2 ]    dy

This is the formula for finding the volume using washers when the region is rotated about the y-axis.

Example 3: Find the volume of the solid that results when the region bounded by the curve x = y^2 and the
curve x = y^3 , from y = 0 to y = 1, is revolved about the y-axis.

Sketch away.

Because x = y^2 is always on the outside and x = y^3 is always on the inside, you have to evaluate the
integral.

π (y^4  −   y^6 )   dy

You should get the following: