The area under a curve can be determined both using Cartesian plane with rectangular coordinates, and polar coordinates. For instance the polar equation describes a curve. The formula for the area under this polar curve is given by the formula below:
Consider the arc of the polar curve traced as varies from to . If this arc bounds a closed region of the plane, the area of this region is
Consider a circle of radius ; a sector of this circle subtending an angle of will have area
For an infinitesimal change in the angle , from to , the region of the plane swept out by the polar curve is approximately a circular sector, hence has area Integrating this factor over the interval gives the area swept by the curve as which is the desired formula.
What is the area of a circle with radius ?
A circle of radius is swept out by the polar equation as varies from to . Thus, the area of the circle is
A rose curve is defined by the polar equation . This curve has three "petals"; what is the area bounded by one of the petals?
As varies from to , the curve traces out one petal. Thus, the area of a petal is