Answer to Question #269033 in Calculus for Faru

Question

4.Find the area bounded by the cardiode (i) r=a(1+cos theta) , (ii) r=a( 1-cos theta) .

Answer 1

i)

$S=\frac{1}{2}\intop r^2(\theta)d\theta=a^2\intop^{\pi}_0(1+cos \theta)^2d\theta=$

$=a^2(\pi+(\frac{cos\theta sin\theta+\theta}{2})|^{\pi}_0)=3\pi a^2/2$

ii)

$S=a^2\intop^{\pi}_0(1-cos \theta)^2d\theta==a^2(\pi+(\frac{cos\theta sin\theta+x}{2})|^{\pi}_0)=3\pi a^2/2$