Exercise 1917Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Area of a region using line integral

#line integral#area#polar

Statement

Use $\frac12\oint(x\,dy-y\,dx)$ to compute the area of the region enclosed by $r=1+\cos\theta$ (cardioid).

Hint

$x\,dy-y\,dx=r^2\,d\theta$ for polar curves.

Show solutionHide solution

Polar form. $x=r\cos\theta$ , $y=r\sin\theta$ ; direct computation shows $x\,dy-y\,dx=r^2\,d\theta$ .

Integral. $\text{Area}=\frac12\int_0^{2\pi}(1+\cos\theta)^2\,d\theta$ .

Compute. $(1+\cos\theta)^2=1+2\cos\theta+\cos^2\theta$ ; integral over $[0,2\pi]$ : $2\pi+0+\pi=3\pi$ .

Result. Area $=\frac12\cdot 3\pi=\frac{3\pi}{2}$ .

Conclusion. Area of cardioid $=3\pi/2$ .