Example 2: Area of a Circle in Polar Coordinates

Find the area of the region inside the circle r = sin θ (Figure 7.9.7).

Figure 7.9.7

The point (r, θ) goes around the circle once when 0 ≤ θ ≤ π with r positive, and again when π ≤ θ ≤ 2π with r negative. The theorem says that we will get the correct area if we take either 0 and π, or π and 2π, as the limits of integration. Thus

A == ¼(π - 0) = π/4.

Alternatively,

A == ¼(2π - 7t) = π/4.

Since the curve is a circle of radius ½, our answer π/4 agrees with the usual formula A = πr².

Integrating from 0 to 2π would count the area twice and give the wrong answer.