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Mathematics LibreTexts

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 10.3.1. Recall that the area of a sector of a circle is αr2/2, where α is the angle subtended by the sector. If the curve is given by r=f(θ), and the angle subtended by a small sector is Δθ, the area is (Δθ)(f(θ))2/2. Thus we approximate the total area as

n1i=012f(θi)2Δθ.

In the limit this becomes

ba12f(θ)2dθ.

Example 10.3.1: Area inside a cardiod

We find the area inside the cardioid r=1+cosθ.

imageedit_1_7403075009.png

Figure 10.3.1: Approximating area by sectors of circles.

Solution

2π012(1+cosθ)2dθ=122π01+2cosθ+cos2θdθ=12(θ+2sinθ+θ2+sin2θ4)|2π0=3π2.

Example 10.3.2: area between circles

We find the area between the circles r=2 and r=4sinθ, as shown in 10.3.2.

imageedit_10_8102835346.png

Figure 10.3.2: An area between curves.
Solution

The two curves intersect where 2=4sinθ, or sinθ=1/2, so θ=π/6 or 5π/6. The area we want is then 125π/6π/616sin2θ4dθ=43π+23.

This example makes the process appear more straightforward than it is. Because points have many different representations in polar coordinates, it is not always so easy to identify points of intersection.

Example 10.3.3: area between curves

We find the shaded area in the first graph of 10.3.3 as the difference of the other two shaded areas. The cardioid is r=1+sinθ and the circle is r=3sinθ.

imageedit_11_7578629110.png

Figure 10.3.3: An area between curves.

Solution

We attempt to find points of intersection:

1+sinθ=3sinθ1=2sinθ1/2=sinθ.

This has solutions θ=π/6 and 5π/6; π/6 corresponds to the intersection in the first quadrant that we need. Note that no solution of this equation corresponds to the intersection point at the origin, but fortunately that one is obvious. The cardioid goes through the origin when θ=π/2; the circle goes through the origin at multiples of π, starting with 0.

Now the larger region has area

12π/6π/2(1+sinθ)2dθ=π29163

and the smaller has area

12π/60(3sinθ)2dθ=3π89163 so the area we seek is π/8.

Contributors


This page titled 10.3: Areas in Polar Coordinates is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David Guichard via source content that was edited to the style and standards of the LibreTexts platform.

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