AP Calculus BC 8.9: Solids of Revolution

\begin{tabular}{|l|l|l|}
\hline \multirow[t]{2}{*}{Volume $=\int_{0}^{2} \pi\left(c x \sqrt{4-x^{2}}\right)^{2} d x=\pi c^{2} \int_{0}^{2} x^{2}\left(4-x^{2}\right) d x$} & Form of the integrand & 1 point \\
\hline & Limits and constant & 1 point \\
\hline $=\pi c^{2} \int_{0}^{2}\left(4 x^{2}-x^{4}\right) d x=\pi c^{2}\left(\frac{4}{3} x^{3}-\left.\frac{1}{5} x^{5}\right|_{0} ^{2}\right)$ & Antiderivative & 1 point \\
\hline $$\begin{aligned} & =\pi c^{2}\left(\frac{32}{3}-\frac{32}{5}\right)=\frac{64 \pi c^{2}}{15} \\ & \frac{64 \pi c^{2}}{15}=2 \pi \Rightarrow c^{2}=\frac{15}{32} \Rightarrow c=\sqrt{\frac{15}{32}} \end{aligned}$$ & Answer & 1 point \\
\hline
\end{tabular}

Scoring notes:

- The first point is earned for presenting an integrand of the form $A\left(x \sqrt{4-x^{2}}\right)^{2}$ in a definite integral

with any limits of integration (numeric or alphanumeric) and any nonzero constant A. Mishandling

the c will result in the response being ineligible for the fourth point.

- The second point can be earned without the first point. The second point is earned for the limits of

integration, x=0 and x=2, and the constant $\pi$ ( but not for $2 \pi$ ) as part of an integral with a

correct or incorrect integrand.

- If an indefinite integral is presented with the correct constant $\pi$, the second point can be earned if

the antiderivative (correct or incorrect) is evaluated using the correct limits of integration.

- A response that presents $2=\int_{0}^{2}\left(c x \sqrt{4-x^{2}}\right)^{2} d x$ earns the first and second points.

- The third point is earned for presenting a correct antiderivative of the presented integrand of the

form $A\left(x \sqrt{4-x^{2}}\right)^{2}$ for any nonzero A. If there are subsequent errors in simplification of the

antiderivative, linkage errors, or sign errors, the response will not earn the fourth point.

- The fourth point cannot be earned without the third point. The fourth point is earned only for the

correct answer. The expression does not need to be simplified to earn the fourth point.