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\textbf{Name:}$\underline{\hspace{3in}}$\\
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{\bf
Math 115: Applied Calculus

Module 5 Exam}

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\begin{tabular}{| c | c | c |}
\hline \ \ Problem\ \  & \ \ \ Points\ \ \  & \ \ \ Score\ \ \ \\
\hline 1 & 20 & \\
\hline 2 & 20 & \\
\hline 3 & 10 & \\
\hline 4 & 20 & \\
\hline 5 & 15 & \\
\hline 6 & 15 & \\
\hline Total & 100 & \\
\hline
\end{tabular}
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\noindent \textbf{SHOW ALL WORK.} May the force be with you.

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% XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX  Problem 1

\item (20 points)
Approximate the area under the curve {\LARGE $f(x) = 2x^2$} from $x = 2$
to $x = 3$ using each of the following methods.

{\bf a.} (10 points) A Right Riemann Sum with $n = 5$.

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{\bf b.} (10 points) A Midpoint Riemann Sum with $n = 5$.

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% XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX  Problem 2

\item (20 points)
Determine the exact value of each definite integral.

{\bf a.} (10 points)

{\LARGE $\displaystyle\int_1^2 x^3 + 4\sqrt{x} \, dx$}

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{\bf c.} (10 points)

{\LARGE $\displaystyle\int_{0}^\pi 2e^{-x} +2\cos(2.5x)\, dx$}







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% XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX  Problem 3

\item (10 points)
Determine the average value of the function {\LARGE $f(x) = \dfrac{1}{x}$} as $x$ ranges from $x = 1$ to $x = 5$.




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% XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX  Problem 4

\item (20 points)
What is the area between the curves $y = x^3$ and $y = 4x$ as
$x$ ranges from $-2$ to $2$?   You may {\large {\bf not}} use the graphical feature on your
calculator for this question.

{\it Hint: Find the (three) intersection points
of these functions and then plug in $x$-values between the
intersection points to determine which function is larger on that
interval.}

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% XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX  Problem 5

\item (15 points)
What is the volume of the solid formed by rotating $y = 2x - 3$
as $x$ ranges from $0$ to $2$ about the $x$-axis?

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% XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX  Problem 6

\item (15 points)
Compute the double integral.
\[
    {\int_0^1 \left( \int_0^3 x^2y - e^x \, dy \right) dx}
\]

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