MATH 32B Discussion Week 1
September 30, 2025
Problem 1. Compute the following double integrals:
Z Z
Z Z
3
x dA
ex sin ydA
[−4,4]×[0,5]
Z Z
e
[0,1]×[1,2]
[0,2]×[0,π/4]
3x+4y
Z Z
√
dA
[0,4]×[0,5]
Z Z
x log(y)dA
[0,1]×[1,2]
1
dA
x+y
Note: Here I wrote ”log x” meaning its in base e, some of you might be more familiar with
the notation ln(x)
1
Problem 2. Compute the following double integrals using Fubini’s Theorem :
Z Z
Z Z
Z Z
p
x
dxdy
y 1 + xydA
xexy dA
1
+
xy
[0,1]×[0,1]
[0,1]×[0,1]
[0,1]×[0,1]
2
Problem 3. Show that
Z ∞
2
e−t dt =
√
π
−∞
by considering
Z ∞Z ∞
0
2
e−x(1+y ) dxdy
0
Hint. At some point you might need the property that
Z ∞ −x
Z ∞
e
2
√ dx = 2
e−x dx, Try and show this!
x
0
0
3
Problem 4. Show that
Z 1Z 1
0
0
x2 − y 2
dxdy ̸=
(x2 + y 2 )2
Z 1Z 1
0
0
Why doesn’t the Fubini Theorem you learned apply?
4
x2 − y 2
dydx
(x2 + y 2 )2