MATH 251 – LECTURE 16
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 13.1–4
webAssign: 13.1–3, due 2/29 11:55 p.m.
Next week: 13.4–6,8
webAssign: 13.4-6, opens 2/29 12 a.m.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Double integrals over more general domains
Regions of type I:
D = {(x, y) | a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}.
Regions of type II:
D = {(x, y) | h1(y) ≤ x ≤ h2(y), c ≤ y ≤ d}.
ZZ
Z
b
g2 (x)
Z
f (x, y) dA =
f (x, y) dy dx.
D
a
Z
ZZ
g1 (x)
d
Z
h2 (y)
f (x, y) dA =
D
!
!
f (x, y) dx dy.
c
h1 (y)
Double integrals over more general domains
Both type I and II
Neither type I nor II
Type I only
Type II only
Double integrals over more general domains
What if the domain D is neither of type I nor of type II?
Exercise 1. Subdivide the region {1 ≤ x2 + y 2 ≤ 4} into regions of either type I or type II.
Double integrals over more general domains
Exercise 2. Consider the type I region D = {0 ≤ x ≤ 1, x ≤ y ≤ 1}. Express D as a type II region.
Double integrals over more general domains
2
Exercise 3. Compute the double integral of f (x, y) = ey over the type I region D = {0 ≤ x ≤ 1, x ≤ y ≤ 1}.
Polar coordinates
Change of variables
Exercise 4. Compute the integral
Z
2
3
1
dx.
x ln(x)
Polar coordinates - Change of variables
Polar coordinates
Exercise 5. Integrate the function f (x, y) = x over the region D = {1 ≤ x2 + y 2 ≤ 4}.
Polar coordinates
Exercise 6. Integrate the function f (x, y) = x2 y over the region D = {x ≥ 0, y ≥ 0, x2 + y 2 ≤ 1}.