AP Calculus BC 8.7: Cross-Section Volumes

Volume $=\int_{1}^{5} x e^{x / 5} d x$

Definite integral

1 point

Using integration by parts,
$$\begin{array}{ll} u=x d v=e^{x / 5} d x d u=d x v=5 e^{x / 5} \end{array}$$

u and d v

1 point

$$\begin{aligned} \int x e^{x / 5} d x=5 x e^{x / 5}-\int 5 e^{x / 5} d x =5 x e^{x / 5}-25 e^{x / 5}+C =5 e^{x / 5}(x-5)+C \end{aligned}$$$$\begin{aligned} \int x e^{x / 5} d x =5 x e^{x / 5}-\int 5 e^{x / 5} d x \end{aligned}$$

1 point

$$\begin{aligned} \text { Volume }=\left.5 e^{x / 5}(x-5)\right|_{1} ^{5} =5 e(0)-5 e^{1 / 5}(-4)=20 e^{1 / 5} \end{aligned}$$

Answer

1 point

Scoring notes:

- The first point is earned for $c \int_{1}^{5} x e^{x / 5} d x$, where $c \neq 0$. Errors of $c \neq 1$, for example $c=\pi$, will

not earn the fourth point.

- Incorrect integrals that require integration by parts are still eligible for the second and third points.

Both of these points will be earned with at least one correct application of integration by parts.

- The second point will be earned with an implied u and d v in the presence of $5 x e^{x / 5}-\int 5 e^{x / 5} d x$.

- The tabular method may be used to show integration by parts. In this case, the second point is earned

by having columns (labeled or unlabeled) that begin with x and $e^{x / 5}$. The third point is earned for

either $5 x e^{x / 5}-\int 5 e^{x / 5} d x$ or $5 x e^{x / 5}-25 e^{x / 5}$.

- Limits of integration may be present, omitted, or partially present in the work for the second and

third points.

- The fourth point is earned only for the correct answer.

Total for part (b)

4 points