Recipe

Find the area enclosed by two overlapping polar curves

Use when: the region shared by, or between, two polar curves is needed, not just one curve's own interior

Goal: the total shared or between-curve area, built from separate sector-area integrals over each angle range where a different curve is the boundary

Taught in Area Between Polar Curves and Finding Intersections · extends Find the area enclosed by a polar curve

1. 1

   Set r_1(\theta)=r_2(\theta) and solve for the intersections that share the same theta. Then separately check whether either curve passes through the pole, since a shared pole crossing can happen at a different theta for each curve.

   Example

   Circle r=1 and cardioid r=1+\cos\theta: setting them equal, 1=1+\cos\theta\implies\cos\theta=0\implies\theta=\dfrac\pi2,\dfrac{3\pi}2. The circle r=1 never reaches the pole, so no extra pole intersection exists here.

   • both curves reach r=0 at some (possibly different) theta — the pole is a shared intersection point even though the algebra above won't reveal it; add it to the list by hand.
   • at most one curve ever reaches r=0 — the pole isn't a shared intersection, and the algebraic solutions above are the complete list.

   Why: setting r_1(\theta)=r_2(\theta) only finds crossings that happen at the same theta for both curves, and misses a point each curve reaches using a different, equally valid theta, which happens most often at the pole.

2. 2

   Test one theta-value in each range between consecutive intersections to see which curve has the smaller r.

   Example

   For \theta\in\left(-\dfrac\pi2,\dfrac\pi2\right): at \theta=0, circle gives r=1, cardioid gives r=2, so the circle is smaller there. For \theta\in\left(\dfrac\pi2,\dfrac{3\pi}2\right): at \theta=\pi, circle gives r=1, cardioid gives r=0, so the cardioid is smaller there instead.

   Why: which curve bounds the region switches at every intersection, exactly the way "top" and "bottom" can switch in a rectangular-coordinates area-between-curves problem.

3. 3

   Write a separate \tfrac12\int r^2\,d\theta integral for each angle range identified in Step 2, using whichever curve is the boundary on that range.

   Example

   A = \frac12\int_{-\pi/2}^{\pi/2}(1)^2\,d\theta + \frac12\int_{\pi/2}^{3\pi/2}(1+\cos\theta)^2\,d\theta

   Why: this is exactly the single-curve recipe above, applied piecewise, once per range where a different curve is the active boundary.

4. 4

   Evaluate each integral from Step 3 separately, then sum the results.

   Example

   \frac12\int_{-\pi/2}^{\pi/2}1\,d\theta = \frac\pi2, \qquad \frac12\int_{\pi/2}^{3\pi/2}(1+\cos\theta)^2\,d\theta = \frac{3\pi}{4}-2\boxed{A = \frac\pi2+\frac{3\pi}4-2 = \frac{5\pi}{4}-2}

   Why: adding the pieces is only valid once each range has been matched to the correct boundary curve in Step 2; adding the wrong pairing produces a completely different number.