MATH 396 PROBLEMS 2

IGOR KRIZ

Regular problems:

1. Calculate the derivative (by t) of the function

F (t) = ∫_{1}^{2} \frac{e^{tx}}{x} dx.

[Do not try to integrate directly.]

2. Calculate:

∫_{[0,1]×[0,1]} \frac{dxdy}{\sqrt[3]{xy}} .

3. Using Fubini’s theorem, calculate

∫_{[0,1]×[0,1]} \frac{dxdy}{x + y} .

4. Using polar coordinates, calculate

∫_{B} \frac{xy}{x^{2} + y^{2}} dxdy

where B is the solid circle of radius 1 with center in the origin.

Challenge problem:

5. (a) Define 3-dimensional coordinates of a point (x, y, z) as (r, α, β) where (r, α) are the 2-dimensional polar coordinates of (x, y) and β is the angle of the vector (x, y, z) with the plane determined by the x and y axes. Find the Jacobian of the map expressing the 3-dimensional Cartesian coordinates x, y, z in terms of the polar coordinates r, α, β.

(b) Using (a), calculate:

∫_{B} \frac{dxdydz}{(\sqrt{x^{2} + y^{2} + z^{2}})^{t}}

where B is the set of all points of distance ≤ 1 from 0, and t is a parameter. (c) Can you generalize (a) to dimension n?

(d) For which values of t does

∫_{B} \frac{dx_{1}...dx_{n}}{(\sqrt{x_{1}^{2} + ... + x_{n}^{2}})^{t}}

converge where B is the ball in ℝ^{n} with radius 1 and center in the origin? [half credit for a correct conjecture...]