Area in Polar Coordinates

One of the most elegant applications of polar coordinates is computing the area enclosed by a polar curve. The formula differs from the Cartesian one and is a key part of the AQA Further Mathematics specification.

---

1 The Area Formula

Derivation

Consider a polar curve r = f(theta). A thin "sector" from angle theta to theta + delta theta has:

• Radius approximately r = f(theta)
• Angle delta theta

The area of this sector is approximately (1/2) r^2 delta theta (using the sector area formula).

Summing all such sectors and taking the limit:

A = (1/2) integral from alpha to beta of r^2 d theta

where alpha and beta are the angular limits.

Key Point: This gives the area swept by the radius vector from theta = alpha to theta = beta. It is the area between the curve and the pole.

---

2 Worked Example 1 -- Circle

Find the area enclosed by r = a.

Solution:

A = (1/2) integral from 0 to 2pi of a^2 d theta = (1/2) a^2 [theta] from 0 to 2pi = (1/2) a^2 (2pi) = pi a^2