Lecture 17 (Oct. 23)
Double Integrals in Polar Coordinates (reading: 15.4)
Recall: polar coordinates, and why they are useful:
How to compute a double integral using polar coordinates:
So if f (x, y) is continuous on the “polar rectangle” R given by a  r  b, ↵  ✓ 
, we have
ZZ
Z Z b
f (x, y)dA =
f (r cos(✓), r sin(✓))rdrd✓.
R
↵
a
Key point: dA = dxdy is replaced by rdrd✓.
10
More generally, if f (x, y) is continuous on the polar region
D := {(r, ✓) | ↵  ✓  , h1 (✓)  r  h2 (✓)},
then
ZZ
f (x, y)dA =
D
Z
↵
Z
h2 (✓)
f (r cos(✓), r sin(✓))rdrd✓.
h1 (✓)
Example: find the volume inside the sphere x2 + y 2 + z 2 = 16 and outside the cylinder
x2 + y 2 = 4.
11
Example: find the area of one “loop” of r = cos(3✓).
12