Area Between Polar Curves

Calculus 2 · Axiom Academy

LESSON Area Between Polar Curves Outer minus inner, swept as sectors — find where two polar curves bound a region and integrate the gap. 1. Outer Sector Minus Inner Sector A polar region is swept by a ray turning from to . At each angle the ray spans from the inner radius out to the outer radius — a thin sector of width . Watch the wedge sweep: the area it adds is the big sector minus the small one. This comes straight from the basic polar-area formula : the outer curve sweeps one area, the inner curve sweeps a smaller one, and the region between is the difference. The factor of is there because each slice is a circular sector , not a rectangle. The limits come from the intersections . Set and solve for . In the animation the cardioid and the circle r_2=1 are plotted from their real equations, and the crossing angles are marked where the two radii are genuinely equal. Keep the solutions in range (often ). Test a sample in each region to see which curve is outer ( r_1 ) and which is inner ( r_2 ). Find the area inside the cardioid and outside the circle r_2=2 . Press play: the ray sweeps the region where the cardioid is genuinely outside the circle, and the swept sector area accumulates as to its final value. Which is outer: at , , so the cardioid is outside on . Set up (use symmetry): integrate and double. Evaluate the definite integral. 4. When the Curves Trade Places

This is the written version of the interactive lesson above. See the full Calculus 2 course.