MATH 182 $\;$TEST II March 10, 1998 Name

SHOW ALL WORK Instructor/Section

1.

Find the area of the region bounded by $\quad y=x+5, \quad y=2, \quad y=-1, \quad y^{2}=x$.

2.

Express as an integral the volume of the solid obtained by rotating the region bounded by $\quad y=x^{3}, \quad y=8, \;\;$ and $\;\;x=0 \quad$ about the x -axis. $\quad$ Do not evaluate the integral.

3.

Express as an integral the volume of the solid whose base is the region in the first quadrant bounded by $\quad x=0, \quad y=x, \quad y=1 \quad$ and whose cross sections perpendicular to the x -axis are squares. Do not evaluate the integral.

4.

Express as an integral the arc length of the curve $\;\; x=\sin(2t), \quad y=-\cos(2t), \quad -\frac{\pi}{2} \leq t \leq \frac{\pi}{2}$. Then evaluate the integral.

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5.

A direction field is given below. Which of the following represents its differential equation?

Answer(s):

6.

(a)

Show that every member of the family of functions $\; y=ce^{x^{3}/3} \;$ is a solution of the differential equation $\; y'=x^{2}y. \;$

(b)

Find a solution of the differential equation $y'=x^{2}y \;$ that satisfies the initial condition $\; y(0)=8.$

7.

Let f be the function whose graph is shown below. Find $\;\;{\displaystyle \int_{0}^{6} f(x)\;dx}\;\;$ and the average value of f .

8.

The cylindrical tank shown is full of water. Approximate with a Riemann sum the required work to pump the water out of the tank. Then express the work as an integral. Do not evaluate the integral. Recall that the density of water is $62.5 \; \textnormal{lbs/ft}^{3}$.