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Double Integrals in Polar Coordinates

A point with polar coordinates (θ, r) has rectangular coordinates

(x, y) = (r cos θ, r sin θ).

DEFINITION

A polar region is a region D in the (x, y) plane given by polar coordinate inequalities

α ≤ 0 ≤ β, a(θ) ≤ r ≤ b(θ),

where a(θ) and b(θ) are continuous. To avoid overlaps, we also require that for all (0, r) in D,

0 ≤0 ≤2π and 0 ≤ r.

The last requirement means that the limits α and β are between 0 and 2π, while the limits a(θ) and b(θ) are ≥ 0. Figure 12.5.1 shows a polar region.

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Figure 12.5.1

The simplest polar regions are the polar rectangles

α ≤ θ ≤β, a ≤ r ≤ b.

We see in Figure 12.5.2 that the θ boundaries are radii and the r boundaries are circular arcs.

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Figure 12.5.2

The polar rectangle

α≤θ ≤ β, 0≤ r ≤ b

is a sector of a circle of radius b (Figure 12.5.3(a)).

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Figure 12.5.3 (a) sector: α ≤ θ ≤ β, 0≤ r ≤b

The polar rectangle

0 ≤ θ ≤ 2π, 0 ≤ r ≤ b

is a whole circle of radius b (Figure 12.5.3(b)).

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Figure 12.5.3 (b) circle: 0 ≤ θ ≤ 2π, 0 ≤ r ≤ b

Less trivial examples of polar regions are the circle with diameter from (0,0) to (0, b),

0 ≤ θ ≤π, 0 ≤ r ≤ b sin θ,

and the cardioid 

0 ≤ θ ≤ 2π, 0 ≤ r ≤ 1 + cos θ.

Both of these regions are shown in Figure 12.5.4.

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Figure 12.5.4 (a) circle: 0 ≤ θ ≤ ir, 0 ≤ r ≤ b sin θ

(b) cardioid:

0 ≤ θ ≤ 2π, 0 ≤ r ≤ 1 + cos θ

 

We shall use the Infinite Sum Theorem to get a formula for the double integral over a polar region. In the proof we take for ΔD an infinitely small polar rectangle.


Last Update: 2010-11-25