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# 10.3: Areas in Polar Coordinates

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We can use the equation of a curve in polar coordinates to compute some areas bounded by such curves. The basic approach is the same as with any application of integration: find an approximation that approaches the true value. For areas in rectangular coordinates, we approximated the region using rectangles; in polar coordinates, we use sectors of circles, as depicted in $$\PageIndex{1}$$. Recall that the area of a sector of a circle is $$\alpha r^2/2$$, where $$\alpha$$ is the angle subtended by the sector. If the curve is given by $$r=f(\theta)$$, and the angle subtended by a small sector is $$\Delta\theta$$, the area is $$(\Delta\theta)(f(\theta))^2/2$$. Thus we approximate the total area as

$$\sum_{i=0}^{n-1} {1\over 2} f(\theta_i)^2\;\Delta\theta.$$

In the limit this becomes

$$\int_a^b {1\over 2} f(\theta)^2\;d\theta.$$

Example $$\PageIndex{1}$$: Area inside a cardiod

We find the area inside the cardioid $$r=1+\cos\theta$$.

Figure $$\PageIndex{1}$$: Approximating area by sectors of circles.

Solution

$$\int_0^{2\pi}{1\over 2} (1+\cos\theta)^2\;d\theta= {1\over 2}\int_0^{2\pi} 1+2\cos\theta+\cos^2\theta\;d\theta= {1\over 2}\left.(\theta +2\sin\theta+ {\theta\over2}+{\sin2\theta\over4})\right|_0^{2\pi}={3\pi\over2}.$$

## Contributors

• Integrated by Justin Marshall.