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Chapter 15 – Multiple Integrals15.4 Double Integrals in Polar Coordinates

15.4 Double Integrals in Polar Coordinates

Objectives: Determine how to express

double integrals in polar coordinates

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15.4 Double Integrals in Polar Coordinates 2

Polar Coordinates ReviewRecall 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

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15.4 Double Integrals in Polar Coordinates 3

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

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15.4 Double Integrals in Polar Coordinates 4

Double Integrals in Polar Coordinates

In either case, the description of R in terms of rectangular coordinates is rather complicated but R is easily described by polar coordinates.

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15.4 Double Integrals in Polar Coordinates 5

Polar RectangleThe regions in the first figure are special cases of a polar

rectangle R = {(r, ) | a ≤ r ≤ b, ≤ ≤ }

shown here.

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15.4 Double Integrals in Polar Coordinates 6

Polar RectangleTo compute the double integral

where R is a polar rectangle, we divide:

◦ The interval [a, b] into m subintervals [ri–1, ri] of equal width ∆r = (b – a)/m.

◦ The interval [ ,] into n subintervals [ j–1, j] of equal width ∆ = ( – )/n.

( , )R

f x y dA

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15.4 Double Integrals in Polar Coordinates 7

Polar SubrectangleThe “center” of the polar subrectangle

Rij = {(r, θ) | ri–1 ≤ r ≤ ri, θj–1 ≤ θ ≤ θi}

has polar coordinates

ri* = ½ (ri–1 + ri)

θj* = ½ (θj–1 + θj)

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15.4 Double Integrals in Polar Coordinates 8

Polar RectanglesThe rectangular coordinates of the center of Rij are (ri*

cos θj*, ri* sin θj*).

So, a typical Riemann sum is shown with Equation 1:

* * * *

1 1

* * * * *

1 1

( cos , sin )

( cos , sin )

m n

i j i j ii j

m n

i j i j ii j

f r r A

f r r r r

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15.4 Double Integrals in Polar Coordinates 9

Polar Rectangles If we write g(r, θ) = r f(r cos θ, r sin θ), the Riemann

sum in Equation 1 can be written as:

◦ This is a Riemann sum for the double integral

* *

1 1

( , )m n

i ji j

g r r

( , )b

ag r dr d

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15.4 Double Integrals in Polar Coordinates 10

Polar RectanglesThus, we have:

* * * *

,1 1

* *

,1 1

( , ) lim ( cos , sin )

lim ( , )

( , )

( cos , sin )

m n

i j i j im n

i jR

m n

i jm n

i j

b

a

b

a

V f x y dA f r r A

g r r

g r dr d

f r r r dr d

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15.4 Double Integrals in Polar Coordinates 11

Change to Polar Coordinates

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15.4 Double Integrals in Polar Coordinates 12

Change to Polar CoordinatesFormula 2 says that we convert from rectangular to

polar coordinates in a double integral by:

◦Writing x = r cos θ and y = r sin θ◦Using the appropriate limits of integration

for r and θ◦Replacing dA by dr dθ

Be careful not to forget the additional factor r on the right side of Formula 2.

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15.4 Double Integrals in Polar Coordinates 13

Change to Polar CoordinatesA classical method for remembering

the formula is shown here.

◦ The “infinitesimal” polar rectangle can be thought of as an ordinary rectangle with dimensions r dθ and dr.

◦ So, it has “area” dA = r dr dθ.

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15.4 Double Integrals in Polar Coordinates 14

Example 1Sketch the region whose area is given by

the integral and evaluate the integral.

4cos2

0 0

rdrd

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15.4 Double Integrals in Polar Coordinates 15

Example 2Evaluate the integral by changing to polar

coordinates.

2 2 2 2

, where is the region in the first quadrant that

lies between the circles 4 and 2 .

D

xdA D

x y x y x

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15.4 Double Integrals in Polar Coordinates 16

More Complicated RegionsWhat we have done so far can be extended to the more

complicated type of region shown here.

◦ It’s similar to the type II rectangular regions considered in Section 15.3

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15.4 Double Integrals in Polar Coordinates 17

More Complicated Volumes If f is continuous on a polar region

of the form

D = {(r, θ) | α ≤ θ ≤ β, h1(θ) ≤ r ≤ h2(θ)}

then2

1

( )

( )( , ) ( cos , sin )

h

hD

f x y dA f r r r dr d

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15.4 Double Integrals in Polar Coordinates 18

Volumes and Areas In particular, taking f(x, y) = 1, h1(θ) = 0,

and h2(θ) = h(θ) in the formula, we see that the area of the region D bounded by θ = α, θ = β, and r = h(θ) is:

Which agrees with formula 3 in section 10.4.

( )

0

( )221

2

0

( ) 1

[ ( )]2

h

D

h

A D dA r dr d

rd h d

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Example 3Use a double integral to find the area of

the region.

The region inside the cardiod 1 cos and outside

the circle 3cos .

r

r

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15.4 Double Integrals in Polar Coordinates 20

Example 4 Use polar coordinates to find the volume

of the given solid.2 2Bounded by the paraboloid 1 2 2

and the plane 7 in the first octant.

z x y

z

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Example 5 Evaluate the iterated integral by

converting to polar coordinates.22 2

2 2

0 0

x x

x y dydx

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15.4 Double Integrals in Polar Coordinates 22

More Examples

The video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 2◦Example 3◦Example 4

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DemonstrationsFeel free to explore these demonstrations

below.◦Double Integral for Volume

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