Cracking The Ap Calculus ab Exam 2018

π (y^4  −   y^6 )   dx  =   π

There’s only one more nuance to cover. Sometimes you’ll have to revolve the region about a line instead
of one of the axes. If so, this will affect the radii of the washers; you’ll have to adjust the integral to
reflect the shift. Once you draw a picture, it usually isn’t too hard to see the difference.

Example 4: Find the volume of the solid that results when the area bounded by the curve y = x^2 and the
curve y = 4x is revolved about the line y = −2. Set up but do not evaluate the integral. (This is how the AP
Exam will say it!)

You’re not given the limits of integration here, so you need to find where the two curves intersect by
setting the equations equal to each other.

x^2     =   4x

x^2     −   4x  =   0

x   =   0,  4

These will be our limits of integration. Next, sketch the curve.

Notice that the distance from the axis of revolution is no longer found by just using each equation. Now,

you need to add 2 to each equation to account for the shift in the axis. Thus, the radii are x^2 + 2 and 4x + 2.
This means that we need to evaluate the integral.