Question

What is the area common to the circle $r=a$ and $r=2acos\theta ?$

• A. $0.524a^2$
• B. $0.614a^2$
• C. $1.047a^2$
• D. $1.228a^2$

Answer 1

The correct option is D $1.228a^2$


$r=a=2acos\theta$
$cos\theta =\frac{1}{2}\Rightarrow [\theta =\frac{\pi }{3}]$
$\therefore$ Shaded Area $=2$ Area of upper portion
$=2{\int }_{\theta =0}^{\pi /3}{\int }_{r=a}^{2acos\theta }rd\theta dr$
$=2{\int }_0^{\pi /3}{\left[\frac{r^2}{2}\right]}_a^{2acos\theta }d\theta$
$=a^2{\int }_0^{\pi /3}(4co{s}^2\theta -1)d\theta$
$=a^2{\int }_0^{\pi /3}\left(2\right(1+cos2\theta )-1)d\theta$
$=a^2{\int }_0^{\pi /3}(1+2cos2\theta )d\theta$
$=a^2{[\theta +sin2\theta ]}_0^{\pi /3}$
$=a^2[\frac{\pi }{3}+\frac{\sqrt{3}}{2}]=1.9132a^2$
So common Area $=\pi a^2-1.9132a^2$
$=(3.1428-1.9132)a^2=1.228a^2$