# Integral

## Integral

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

- 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

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:

- 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

Example 4
Find the area of the inner loop of the limacon r = a(1 + 2 cos θ).

Solution