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Theorem 1: Area of a Sector of a Circle
In this section we derive a formula for the area of a region in polar coordinates. Section 6.3 on the length of a curve in rectangular coordinates should be studied before this and the following section. Our starting point for areas in rectangular coordinates was the formula for the area of a rectangle. In polar coordinates our starting point is the formula for the area of a sector of a circle. THEOREM 1 A sector of a circle with radius r and central angle θ has area A = ½r2 θ. An arc of a circle with radius r and central angle θ has length s = rθ.
Figure 7.9.1 PROOF Consider a sector POQ shown in Figure 7.9.1. To simplify notation let 0 be the origin, and put the sector POQ in the first quadrant with P on the x-axis. Then P = (½, 0), Q = (r cos θ, r sin θ). The arc QP has the equation
Integrating by the trigonometric substitution x = r sin φ, we get
Therefore A = ½r2θ. By definition, A = ½rs, so
The next theorem gives the formula for area in polar coordinates.
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, r cos θ ≤ x ≤ r. We see from the figure that
Last Update: 2006-11-25