Natural Science Department – Duy Tan University

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


Suppose that we want to evaluate a double integral:

, where R is one of the regions shown here.

( , )R

f x y dA



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



Polar coordinates*



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*



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*



We convert from rectangular to polar coordinates

in a double integral by:

*Replacing x by rcosθ and y by rsinθ

*Replacing dA by rdrdθ



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



Example*

2

0

1520

0

(7cos 15sin ]

[7cos (1 cos 2 )]

15 157sin sin 2

2 4

15

2

d

d



Example *

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Thank you for your attention

Natural Science Department – Duy Tan University

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