Integration Applied to Polar Coordinates to Find Areas

Imagine a lawn sprinkler spraying water where the reach varies in each direction.

The path traced by the spray can be modeled by a polar curve, where the radius is a function of the angle theta.

To approximate the area watered by this uneven spray, divide the region into narrow radial sections.

Because a full circle has a specific area and a total angle of two pi, a section with angle theta has a corresponding proportional area.

This relationship approximates the area of a small slice, where each small area utilizes the radius at a specific angle and a tiny change in angle.

The bounded area equals the limit of the Riemann sums of these sections, where the total area is evaluated as the definite integral.

So, the area bounded by a polar curve from angle a to b is calculated by integrating one-half the function value squared with respect to theta.

Applying this to the sprinkler, with theta ranging from 0 to two pi, gives the exact area watered by the sprinkler.