EXAMINATION 3

MATH 2331

The point value of each problem is indicated in parentheses. There are 30 points all together. Show your work or otherwise explain what you are doing. Write in complete sentences, especially when stating definitions and theorems. Begin each problem on a new page of your blue book. Put the problems in your blue book in the order given.

1. (3) Let R = [−2, 2] × [−1, 1]. Complete the second, fourth and fifth lines of the following Matlab script to calculate the Riemann sum with m = n = 50 to estimate the value of ∫∫_{R}(2x + x^{2}y) dA. Take the sample points to be the midpoints of the subrectangles.

   dx = 4/50; dy = 2/50;
   [x,y] = meshgrid(-2+.04:dx:2-.04,
   z = 2*x + x.^2.*y;
   dA =                             % Area of a subrectangle.
   RS =                             % the Riemann sum.

2. Consider ∫∫_{D}(x − 2y) dA, where D is the domain bounded by the lines y = 2x, y = x + 1 and x = 3.

(a) (1) Sketch the region D.

(b) (2) Express the double integral as an interated integral.

(c) (2) Evaluate the iterated integrals in part (b).

3. For ∫_{0}^{1/2} ∫_{v}^{1−v} \frac{1}{1−u^{2}+v^{2}} dudv:

(a) (1) Sketch the region of integration.

(b) (2) Change the order of integration (do not evaluate it).

4. (a) (2) Express as a double integral the area of the paraboloid z = x^{2} + y^{2} that lies

over the disk x^{2} + y^{2} ≤ 9.

(b) (2) Change to polar coordinates and write the integral as an iterated integral with appropriate limits.

(c) (1) Evaluate it.

(The exam continues on the back of this page)

Date: Friday, April 9, 1999.

5. (3) Find the Jacobian of the transformation

x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ

6. Let T be the tetrahedron bounded by the plane x + 2y + z = 2 and the coordinate planes.

(a) (2) Sketch the region T in xyz-space.

(b) (1) Sketch the face of T in the xy-plane.

(c) (1) Express the volume of T as a triple integral.

(d) (2) Use Fubini’s Theorem to write the triple integral as a triply iterated itegral with appropriate limits. Do not evaluate it.

7. Consider the force field F(x, y) = ⟨x, −xy⟩ in R^{2}.

(a) (2) Sketch the line segment from the point P = (1, 0) to the point Q = (0, 1) and the vector F at the point midway between P and Q.

(b) (3) Find the work done by this force field in moving a particle along the line from P to Q.