Lecture 16: Double Integrals in Polar Coordinates
- §12.3
April 10, 2012 (Tue)
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Review: One Property of Double Integrals
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RR
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Example: Find the area of the region R bounded by y = x
and y = x 2 in the first quadrant.
R
1 dA = A(R) = Area of R
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Review: Polar Coordinates
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In polar coordinates, points are identified by their distance r
from the origin (or pole) and their angle θ from the horizontal
right axis.
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To convert from polar to Cartesian coordinates:
x = r cos θ,
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y = r sin θ
To convert from Cartesian to polar coordinates:
p
y
r = x 2 + y 2 , θ = arctan
x
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Reference: Graphing Polar Functions with Matlab
Three techniques can be used to graph polar equations of the form
r = f (θ): Matlab’s ezplot or polar command. The other is to use
the polar to Cartesian transformations, x = r cos θ and y = r sin θ,
and Matlab’s plot command.
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Matlab’s ezpolar command: Sketch the graph of r = θ.
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Matlab’s Polar Command: Sketch the graph of r = sin 2θ.
1. ezpolar(’theta’) or ezpolar(’theta’, [0, 2*pi])
1. theta = linspace(0, 2*pi);
2. r = sin(2*theta);
3. polar(theta, r)
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Matlab’s Plot Command: Sketch the graph of r = cos 5θ.
1.
2.
3.
4.
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theta = linspace(0, 2*pi);
r = cos(5*theta);
x=r.*cos(theta); y=r.*sin(theta);
plot(x,y)
Some special polar functions: r = a + b cos θ, r 2 = a2 sin nθ
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Double integrals in polar coordinates
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Area(R) =
RR
R
1 dA =
RR
R
r dr dθ.
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Double integrals in polar coordinates
RR
Area(R) =
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Example Find the area enclosed by one loop of the
four-leaves rose r = cos 2θ.
R
1 dA =
RR
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R
r dr dθ.
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Double integrals in polar coordinates
RR
Area(R) =
I
Example Find the area enclosed by one loop of the
four-leaves rose r = cos 2θ.
Double Integral of f (r , θ) over R in polar coordinates
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R
1 dA =
RR
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Z Z
r dr dθ.
Z Z
Z
f (r , θ) dA =
R
R
β
Z
g2 (θ)
f (r , θ) r dr dθ =
R
f (r , θ) r dr dθ
α
g1 (θ)
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Change to Polar Coordinates in a Double Integral
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If f is continuous on a polar rectangle R given by
0 ≤ a ≤ r ≤ b and α ≤ θ ≤ β, then
Z Z
Z
β
b
Z
f (x, y ) dA =
R
f (r cos θ, r sin θ) r dr dθ
α
a
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Change to Polar Coordinates in a Double Integral
I
If f is continuous on a polar rectangle R given by
0 ≤ a ≤ r ≤ b and α ≤ θ ≤ β, then
Z Z
Z
β
b
Z
f (x, y ) dA =
R
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f (r cos θ, r sin θ) r dr dθ
α
a
RR
2
Example Evaluate
R (3x + 4y ) dA, where R is the region
in the upper half-plane bounded by the circles x 2 + y 2 = 1
and x 2 + y 2 = 4. (Volume under the surface)
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Change to Polar Coordinates in a Double Integral
I
If f is continuous on a polar rectangle R given by
0 ≤ a ≤ r ≤ b and α ≤ θ ≤ β, then
Z Z
Z
β
b
Z
f (x, y ) dA =
R
I
I
f (r cos θ, r sin θ) r dr dθ
α
a
RR
2
Example Evaluate
R (3x + 4y ) dA, where R is the region
in the upper half-plane bounded by the circles x 2 + y 2 = 1
and x 2 + y 2 = 4. (Volume under the surface)
RR
√ 1
Example Evaluate
dxdy , where R is the disk
R
of diameter 1 centered at
1−x 2 −y 2
( 12 , 0).
Lecture 16: Double Integrals in Polar Coordinates - §12.3
Change to Polar Coordinates in a Double Integral
I
If f is continuous on a polar rectangle R given by
0 ≤ a ≤ r ≤ b and α ≤ θ ≤ β, then
Z Z
Z
β
b
Z
f (x, y ) dA =
R
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I
a
RR
2
Example Evaluate
R (3x + 4y ) dA, where R is the region
in the upper half-plane bounded by the circles x 2 + y 2 = 1
and x 2 + y 2 = 4. (Volume under the surface)
RR
√ 1
Example Evaluate
dxdy , where R is the disk
R
of diameter 1 centered at
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f (r cos θ, r sin θ) r dr dθ
α
1−x 2 −y 2
( 12 , 0).
Read Example 2 and 4 in §12.3
Lecture 16: Double Integrals in Polar Coordinates - §12.3