Double Integrals in Polar Coordinates — Question 3

Question 3 – Solution

Strategy. Integrate $1$ with radial bounds from the pole to the cardioid over one full turn.

Step 1: Bounds Since $1+\cos\theta\ge 0$, one traversal is $0\le\theta\le 2\pi$, $0\le r\le 1+\cos\theta$.

Step 2: Area integral $$A=\int_0^{2\pi}\int_0^{1+\cos\theta}r\,dr\,d\theta =\frac 12\int_0^{2\pi}(1+\cos\theta)^2d\theta.$$

Step 3: Evaluate Over a full period, $\int\cos\theta=0$ and $\int_0^{2\pi}\cos^2\theta\,d\theta=\pi$. Thus $$A=\frac 12(2\pi+0+\pi)=\boxed{\frac{3\pi}{2}}.$$ The radial bound vanishes only at $\theta=\pi$, so the interval neither omits nor duplicates positive-area portions.