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Exam. I, Math. 120, Fall, 1999

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Your name:

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\textbf{Instructions}:  The exam. consists of 12 problems.  There is
space to work beside each problem.  You should make your steps clear and
readable.  Most of the problems offer multiple-choice answers.  You
should use the answers to check yourself.  It is important to show your
work.  Partial credit may be given for solutions that are set up
correctly, even though the final answer is incorrect, and full credit may
not be given for a correct answer that is not supported by correct work.

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Please remember your obligations under the honor code.

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The spaces below are \emph{not} for answers---but for recording scores.

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1.

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2.

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3.

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4.

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5.

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6.

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7.

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8.

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9.

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10.

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11.

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12.


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1.  What is $\int_{5}^{3} x\,dx$ ?

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(a)  $-10$\hspace{.3in}(b)  $-8$\hspace{.3in}				(c)
$0$\hspace{.3in}
(d)  $8$\hspace{.3in}			 (e)  $10$

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2.  Approximate $\int_{1}^{2} \frac{1}{x}\,dx$, using a Reimann
sum.  Partition the interval into three equal sub-intervals, and evaluate
the function at the left-hand endpoint of each interval.

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(a)  $\frac{2}{3}$\hspace{.3in}	(b)  $\frac{23}{30}$\hspace{.3in}	(c)
$\frac{47}{60}$	\hspace{.3in}(d)
$\frac{4}{5}$ \hspace{.3in}(e)  $\frac{49}{60}$

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3.  If $G(x) = \int_{1}^{2x} \frac{1}{t}\,dt$, then what is
$G'(2)$ ?

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(a)  $\frac{1}{4}$\hspace{.3in}				(b)
$\frac{1}{2}$\hspace{.3in}			 (c)  $1$\hspace{.3in}
			(d)
$2$\hspace{.3in}			 (e)  $4$

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4.  Find $\int_{0}^{\pi} cos^{2}x\,sin\,x\,dx$.

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(a)  $\frac{\pi}{2}$	\hspace{.3in}(b)
$\frac{\pi}{3}$\hspace{.3in} (c)
$1$\hspace{.3in}	(d)  $\frac{1}{3}$\hspace{.3in} (e)  $\frac{2}{3}$

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5.  Find the area of the region bounded by $y=4-x^{2}$ and $y=0$.

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(a)  $\frac{32}{3}$\hspace{.3in}	(b)  $\frac{34}{3}$\hspace{.3in}
(c)
$12$\hspace{.3in} (d)  $\frac{38}{3}$\hspace{.3in}  (e)
$\frac{40}{3}$

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6.  The region bounded by $y=x$, $y=2x$, and $x=1$ is
rotated about the $x$-axis.  What is the volume of the resulting solid of
revolution?

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(a)  $\frac{\pi}{4}$	\hspace{.3in}(b)  $\frac{\pi}{3}$\hspace{.3in}	(c)
$\frac{\pi}{2}\,\pi$\hspace{.3in}	(d)
$\pi$	\hspace{.3in}(e)
$2\pi$


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7.  The region bounded by $y=\frac{1}{x}$, $y=0$, $x=1$, $x=2$ is rotated
about the $y$-axis.  What is the volume of the resulting solid of
revolution?

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(a)  $\frac{\pi}{4}$	\hspace{.3in}(b)  $\frac{\pi}{3}$\hspace{.3in}	(c)
$\frac{\pi}{2}\,\pi$\hspace{.3in}	(d)
$\pi$	\hspace{.3in}(e)
$2\pi$

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8.  Find the volume of the solid having base a unit disk, in
the $xy$-plane, with center $(0,0)$, such that the cross sections
perpendicular to the $x$-axis are squares.

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(a)  $\frac{16}{3}$\hspace{.3in}	(b)  $6$\hspace{.3in} (c)
$\frac{20}{3}$\hspace{.3in}		(d)
$\frac{22}{3}$\hspace{.3in} (e)  $8$

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9.  Holding a certain spring at $1$ meter beyond its natural
length takes a  force of $3$ N.  What is the work done in stretching the
spring from its natural length to $2$ meters beyond?

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(a)  $2$ J\hspace{.3in}	(b)  $3$ J\hspace{.3in}	(c)  $4$
J\hspace{.3in} (d)  $5$ J\hspace{.3in} (e)  $6$ J

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10.  A rope $50$ feet long hangs down the side of a tall
building.  If the rope weighs $1$ lb/ft, what is the work done in lifting
it to the top of the building?

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(a)  $1100$ ft-lbs	(b)  $1150$ ft-lbs	(c)  $1200$ ft-lbs  (d)
$1250$ ft-lbs (e)
$1300$ ft-lbs

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11.  Complete the statement of the following theorem.

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\textbf{Mean-Value Theorem for Integrals}:
Suppose the function $f(x)$ is continuous on the closed interval
$[a,b]$.  Then there is some $c$ in the open interval $(a,b)$ such that

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\[ \int_{a}^{b}f(x)\,dx =\hspace{2in}\]

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12.  Find the average value of the function $f(x) = x^{3}$ on the
interval $[0,2]$.

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(a)  $2$  \hspace{.3in}(b)  $3$\hspace{.3in}  (c)
$4$\hspace{.3in}  (d)
$5$\hspace{.3in}  (e)
$6$

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