Exam Questions - Area bounded by a polar curve

Understanding the area bounded by a polar curve

Polar curves are graphs of functions expressed in polar coordinates.
The area bounded by a polar curve refers to the area enclosed within the curve and its origin (or point of tangency).
The formula used to calculate the area bounded by the polar curve is 1/2 ∫ (from a to b) [r(θ)]² dθ.

You’ll need to be comfortable converting between polar and cartesian coordinates.
Identifying the key features of the curve, such as symmetry, will simplify calculations.
Be sure to find the limits of integration (a and b) which are typically the angles where the curve starts and finishes.
Most areas bounded by a polar curve can be partitioned into smaller, symmetric sections. Calculating the area of a smaller section and multiplying it by the number of times it repeats is another useful strategy.

To calculate the area bounded by the polar curve, first express the curve in polar form, r(θ).
Identify the limits of integration, a and b, based on where the curve begins and finishes.
Evaluate the integral using the formula mentioned above, remembering that the result gives the area of a symmetric section of the total area.
If required, multiply the result by the number of symmetric sections to find the total bounded area.

Always sketch the polar curve to aid understanding of what the area you are calculating represents.
Double-check the accuracy of your polar equation and the limits of angle before integrating.
It’s common for polar curves to have periodic properties or symmetries, use them to your advantage to simplify calculations.
In case the integral cannot be solved exactly, numerical methods on a calculator or a rough approximation using graphs is acceptable.
If your answer is negative, you’ve probably misinterpreted the curve or swapped the limits of integration. Polar areas are always positive, so if you get a negative answer, review your work.