CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-256 Review Exam 3, Spring , 2006

• Your exam shall consist of problems similar to the homework and quiz problems. All homework problems are posted on the course web site. Monday, May 1st will be a review day. Please ask any questions you may have from the sections 8.5, 15.1-15.5, 15.7-15.9. The exam will be held on Wednesday May 3rd in Porter Hall 100. Exams are scheduled to be returned in class on Friday May 5th. Recitation for Thursday May 4th will be cancelled.

• Notes and texts will not be permitted during the exam. Calculators or other electronic devices are not permitted. You will have 50 minutes for the exam, and must stop when told to stop. Please consult the course web page for exam policies - these will be enforced. What follows is a practice exam. This is designed for you to take in 50 minutes. The actual exam will consist of different questions. To fully prepare please practice many homework problems.

1. Evaluate
   $$\iint \limits_R x \cos y \ dA,$$
   where $R$ is the rectangle $\{ (x,y) : 1 \leq x \leq 2, 0 \leq y \leq \pi/2 \}$

2. Evaluate
   $$\iint \limits_D e^{x^2} \ dA,$$
   where $D$ is the triangular region with vertices $(0,0)$, $(2,4)$, and $(2,0)$.

3. Evaluate
   $$\iint \limits_D x \ dA,$$
   where $D$ is the plane region in the first quadrant bounded by the curves $y=x^2$ and $y=-x^2+2x+4$

4. Evaluate
   $$\iint \limits_D \sqrt{4x^2+4y^2} \ dA,$$
   where $D$ is the plane region bounded by the curves $x^2+y^2=4$, $x^2+y^2=1$, $y=0$, and $y=x$.

5. Use the supplied table to find the probability $P(50 \leq X \leq 70 )$ for a continuous random variable $X$ with normal distribution with mean $\mu=58$ and standard deviation $\sigma=16$.

6. Find the probability $P( 4 \leq X+Y )$ for continuous random variables $X$ and $Y$ with probability density function
   $$f(x,y)=\left\{ \begin{array}{lr} 2e^{-x-2y}, & x \geq 0, y \geq 0 \\ 0, & {\rm otherwise} \end{array}\right.$$

7. Evaluate
   $$\iiint \limits_E x \ dV,$$
   where $E$ is bounded by the plane $6x+3y+z=6$ and the coordinate planes $x=0$, $y=0$, and $z=0$.

8. Use spherical coordinates to evaluate
   $$\iiint \limits_E \ dV,$$
   where $E$ is the region in the first octant bounded by the sphere $x^2+y^2+z^2=9$, $y=0$, and the planes $z=0$, $y=0$ and $x=y$.

9. Use the transformation $x=u/3$, $y=v/4$ to evaluate
   $$\iint \limits_D y \ dA,$$
   where $D$ is the region in the first quadrant bounded by the ellipse $9x^2+16y^2=1$.