Mathematical Methods for Social Scientists Math 195 (Sec 45), Autumn 2005 Revision Sheet for the Final Examination

The final will cover all the material covered in this course, and will not place particular emphasis on any part of the syllabus. The following questions cover material not covered in the previous two review sheets. In order to be prepared for the final you should as a bare minimum be able to do all the questions on all three review sheets.

1. (a) State Fubini’s theorem.

(b) Use Fubini’s theorem to evaluate

∫∫_{D} \frac{x}{1 + xy} dA(x, y),

where R = [0, 1] × [0, 1].

(c) Evaluate

∫∫_{D} xye^{x^{2}y} dA(x, y),

where R = [0, 1] × [0, 2].

2. Express the following double integrals as iterated integrals and evaluate them.

(a)

∫∫_{D} \frac{2y}{x^{2} + 1} dA(x, y),

where D = \{(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ \sqrt{x}\}. (b)

∫∫_{D} e^{\frac{x}{y}} dA(x, y),

where D = {(x, y) | 0 ≤ y ≤ 2, y ≤ x ≤ y^{3}}.

3. (a) State the change of variables theorem for double integrals.

(b) Use the identity

(∫_{−∞}^{∞} e^{−x^{2}} dx)^{2} = ∫∫_{R_{2}} e^{−(x^{2}+y^{2})} dA(x, y)

and a change of variables into polar coordinates to prove

∫_{−∞}^{∞} e^{−x^{2}} dx = \sqrt{π}

(c) Evaluate

∫∫_{R} e^{\frac{x−y}{x+y}} dA(x, y),

where R is the trapezoidal region with vertices (2, 0), (3, 0)(0, −3) and (0, −2)