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 ?1 and ?2 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 d?. Note that r is a polar function or r = f(?). See figure above.
– See more at: http://www.mathalino.com/reviewer/integral-calculus/plane-areas-in-polar-coordinates-applications-of-integration#sthash.sOX1NLYx.dpufExample 1
Find the area enclosed by r = 2a sin2?.
 
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 2?.
 
Solution
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The curve is symmetrical with respect to the origin, and occurs only with values of ? from -45° to 45° (-? ? to ? ?).
  
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.dpufExample 3
Find the area inside the cardioid r = a(1 + cos ?) but outside the circle r = a.
 
Solution
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             answerExample 4
Find the area of the inner loop of the limacon r = a(1 + 2 cos ?).
 
Solution
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– See more at: http://www.mathalino.com/reviewer/integral-calculus/example-4-plane-areas-in-polar-coordinates#sthash.Z5nVcZc4.dpufExample 5
Find the area enclosed by four-leaved rose r = a cos 2?.
 
Solution
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? | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
r | a | 0.87a | 0.5a | 0 | -0.5a | -0.87a |…