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\title[Problem Set 7]{Problem Set 7.  Due Thursday, 14 September}

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\maketitle



\noindent {\bf Reading.} {\em Quick Calculus}, pp. 167--170; review 151--185.

\noindent {\bf Supplementary reading.} Simmons, sections 7.2 and 10.7.

\begin{enumerate}

\item (3pts) Compute the following definite integrals.
\begin{enumerate}
\item $\int_0^1x(x^2+2)^3\ dx$
\item $\int_0^1 xe^x\ dx$
\item $\int_0^2\sqrt{4-x}\ dx$
\end{enumerate}

\item (3pts) Find the geometric area of the following functions on the
corresponding interval.

\begin{enumerate}
\item $f(x)=6-3x^2$ on $[0,2]$
\item $f(x)=3x^2-3$ on $[0,3]$
\item $f(x)=9x^2-36$ on $[0,4]$
\end{enumerate}

\item (8pts) Compute the following integrals using integration by
parts.

\begin{enumerate}
\item $\int \frac{\ln(x)}{x}\ dx$
\item $\int x^2e^x\ dx$ (You will have to do the process twice in this 
example.)
\item $\int xe^{ax}\ dx$ for a real number $a$
\item $\int (\ln(x))^2\ dx$
\end{enumerate}


\item (3pts) Find the (geometric) area between the following curves
and the $x$-axis. 

\begin{enumerate}
\item $f(x)=27-3x^2$
\item $f(x)=12-\frac{3}{4}x^2$
\item $f(x)=-2x-\frac{x^2}{2}$
\end{enumerate}

\item (3pts) Find the area of the region bounded by the two curves
given.

\begin{enumerate}
\item $f(x)=\cos(x)$ and $g(x)=\sin(2x)$ on $[0,\frac{\pi}{2}]$
(Hint: $f(x)=g(x)$ when $x=\frac{\pi}{6}$.)
\item $f(x)=x^2-4x$ and $g(x)=2x$
\item $f(x)=7-x^2$ and $g(x)=2$
\end{enumerate}




\end{enumerate}





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