where. Compute g⁰(x).

Z x

2. Letso that (x) = (u)du (i.e., the (x) in Black-Scholes).

• For x > 0, show that (x) = (x).
• Given that lim (x) = 1, use the properties of the integral as well as a substitution

x to show that (x) = 1 (x) (again, assuming x > 0).

3. (a) Under what condition does the following hold?

(b) Evaluate the double integral

ZZ ₂

e^y dA

D

where D = {(x,y) : 0 y 1, 0 x y}.

4. (a) Transform the double integral

into an integral of u and v using the change of variables

u = x + y v = x y

and call the domain in the uv plane S.

• Let D be the trapezoidal region with vertices (1,0), (2,0), (0,2) and (0,1). Find the corresponding region S in the uv plane by evaluating the transformation at the vertices of D and connecting the dots. Sketch both regions.
• Compute the integral found in part (a) over the domain S from part (b).

5. (a) Let D = {(x,y) : 1 x² + y² 9, y 0}. Compute the integral

ZZ ^px² + y² dxdy

D

by changing to polar coordinates. Sketch the domains of integration in both the xy and r (that means r on one axis and on the other) planes.

• Compute the integral

where