10. Parametric Equations & Polar Coordinates

Area & Length in Polar Coordinates

Question #5 (Medium): Area Inside Two Polar Curves

Strategy

First the point of intersection need to be determined by setting the two polar equations equal to each

other. Then based on the symmetry the integral can be simplified. Based on the graph, detect which

polar equation sets the area that lies within both curves. Then use that polar equation to determine te

area base on

Sample Question

Find the area of the region that lies inside both polar curves.

,

Solution

First the points of intersection need to be determined by setting the two polar equations equal to each

other: . Then 4 can be dropped from both sides, then divided by 3 on both

sides. Then: . Divide both sides by , then:

. Let .

Then means

but since ,

and also

, and

. But since the graph is symmetrical in both x-axis and y-axis direction, only a quarter can be set using

integral then multiplied by 4. Since it is the area that lies inside both curves, it is then

upto the point of intersection.

Then the area is:

Therefore the area that lies inside both curves is

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###### Document Summary

Question #5 (medium): area inside two polar curves. First the point of intersection need to be determined by setting the two polar equations equal to each other. Then based on the symmetry the integral can be simplified. Based on the graph, detect which polar equation sets the area that lies within both curves. Then use that polar equation to determine te area base on . Find the area of the region that lies inside both polar curves. First the points of intersection need to be determined by setting the two polar equations equal to each other: . Then 4 can be dropped from both sides, then divided by 3 on both sides. But since the graph is symmetrical in both x-axis and y-axis direction, only a quarter can be set using integral then multiplied by 4. Since it is the area that lies inside both curves, it is then upto the point of intersection.

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