1.5.3—Polar area

The area swept by a polar curve from θ=a to θ=b is A=½∫_a^b r² dθ, provided the interval traces the intended region once.

Square r, so a negative radius contributes positive area. Find intersections and choose limits carefully; if the curve crosses the pole or overlaps itself, split the region to avoid counting it twice.

For r=2cosθ from −π/2 to π/2, A=½∫4cos²θ dθ=π, the area of the corresponding circle.

Do not integrate r rather than r², and do not assume the stated interval is automatically one non-overlapping traversal.