Plane Areas in Polar Coordinates | Applications of Integration
The fundamental equation for finding the area enclosed by a curve whose equation is in polar coordinates is...
Where θ1 and θ2 are the angles made by the bounding radii.
The formula above is based on a sector of a circle with radius r and central angle dθ. Note that r is a polar function or r = f(θ). See figure above.
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Example 1
Find the area enclosed by r = 2a sin2θ.
Solution
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answer
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Example 2
Find the area bounded by the lemniscate of Bernoulli r2 = a2 cos 2θ.
Solution
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The curve is symmetrical with respect to the origin, and occurs only with values of θ from -45° to 45° (-¼ π to ¼ π).
The area in polar coordinates is:
answer
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Example 3
Find the area inside the cardioid r = a(1 + cos θ) but outside the circle r = a.
Solution
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answer
Example 4
Find the area of the inner loop of the limacon r = a(1 + 2 cos θ).
Solution
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answer
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Example 5
Find the area enclosed by four-leaved rose r = a cos 2θ.
Solution
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θ | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
r | a | 0.87a | 0.5a | 0 | -0.5a | -0.87a |...