AP Calculus AB Unit 8.12: Washer Volumes

C Write, but do not evaluate, an integral expression for the volume of the solid generated when the region

R is rotated about the horizontal line y=-2.

- P5 is earned for a definite integral with an integrand of $R^{2}-r^{2}$ or $\left|R^{2}-r^{2}\right|$, where one of $\{R, r\}$

is correct or a difference between g and a nonzero constant, and the other is correct or a difference

between f and a nonzero constant.

- P6 is earned for the integral $\int_{0}^{3}\left((g(x)+2)^{2}-(f(x)+2)^{2}\right) d x, \int_{0}^{3}\left((f(x)+2)^{2}-(g(x)+2)^{2}\right) d x$,

or a mathematically equivalent expression.

- Note P5 and P6 could be earned for a difference of definite integrals.

- A response that presents an integral expression that does not include the constant $\pi$ is eligible for

P5 and P6 but does not earn P7.

- To be eligible for P7, a response must have earned P5.

- A response that reverses the difference of squares must resolve the reversal with either the constant

or the limits of integration AND include the differential to earn P7. For example:

○ A response of $-\pi \int_{0}^{3}\left((f(x)+2)^{2}-(g(x)+2)^{2}\right) d x$ or $\pi \int_{3}^{0}\left((f(x)+2)^{2}-(g(x)+2)^{2}\right) d x$

earns P5, P6, and P7.

○ A response of $\pi \int_{0}^{3}\left((f(x)+2)^{2}-(g(x)+2)^{2}\right) d x$ earns P5 and P6 but does not earn P7.

- A response of only $\pi \int_{0}^{3}\left((g(x)-(-2))^{2}-(f(x)-(-2))^{2}\right) d x$ earns P5, P6, and P7.