Da Nang-09/2013
Natural Science Department – Duy Tan University
Lecturer: Ho Xuan Binh
Double Integrals
in Polar Coordinates
.
We will learn about: How to express double integrals in polar coordinates.
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
Suppose that we want to evaluate a double integral:
, where R is one of the regions shown here.
( , )R
f x y dA
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
In either case, the description of R in terms of rectangular coordinates is rather complicated but R is easily described by polar coordinates.
Recall from this figure that the polar coordinates (r, θ) of a point are
related to the rectangular coordinates (x, y) by the equations:
r2 = x2+y2
x = r cos θ
y = r sin θ
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
Polar coordinates*
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
The regions in the first figure are special cases of a polar rectangle R = {(r, θ) | a ≤ r ≤ b, α ≤ θ ≤ β}
shown here.
Polar rectangle*
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
If f is continuous on a polar rectangle R given by
0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β where 0 ≤ β – α ≤ 2π, then
( , ) ( cos , sin )b
aR
f x y dA f r r r dr d
Formula*
Change to polar coords*
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
We convert from rectangular to polar coordinates
in a double integral by:
*Replacing x by rcosθ and y by rsinθ
*Replacing dA by rdrdθ
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
Example *
where R is the region in the upper half-plane bounded by
the circles x2 + y2 = 1 and x2 + y2 = 4.
Evaluate2(3 4 )
R
x y dA
2
2 2 2
0 1
2 2 3 2
0 1
3 4 2 210
(3 4 )
(3 cos 4 sin )
(3 cos 4 sin )
[ cos sin ]
R
rr
x y dA
r r r dr d
r r dr d
r r d
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
Example*
2
0
1520
0
(7cos 15sin ]
[7cos (1 cos 2 )]
15 157sin sin 2
2 4
15
2
d
d
Double Integrals in Polar Coordinates
Natural Science Department – Duy Tan University
Example *
LOGO
Thank you for your attention
