- Submitted By: aljaycruzado
- Date Submitted: 02/12/2014 5:36 PM
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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.

- See more at: http://www.mathalino.com/reviewer/integral-calculus/plane-areas-in-polar-coordinates-applications-of-integration#sthash.sOX1NLYx.dpuf

Example 1

Find the area enclosed by r = 2a sin2θ.

Solution

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answer

- See more at: http://www.mathalino.com/reviewer/integral-calculus/example-1-plane-areas-in-polar-coordinates#sthash.GrpeLw3b.dpuf

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

- See more at: http://www.mathalino.com/reviewer/integral-calculus/example-2-plane-areas-in-polar-coordinates#sthash.i84y3qJY.dpuf

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

- See more at: http://www.mathalino.com/reviewer/integral-calculus/example-4-plane-areas-in-polar-coordinates#sthash.Z5nVcZc4.dpuf

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 |...