Question

Question #f1f96

Answer 1

`2pi`

Explanation:

The two curves would intersect where `3cos theta = 1+cos theta`. This gives `2 cos theta =1`, or `cos theta= 1/2`. Hence
`theta= pi/3, -pi/3`. The figure showing the common area(shaded in black pen) is given below.

The area would be given by the integral `int_(-pi/3)^(pi/3) {(3cos theta)^2 - (1+cos theta)^2} d theta`

=`int_(-pi/3)^(pi/3) (8cos^2 theta -2cos theta -1)d theta`

=`int_(-pi/3)^(pi/3) (4+4 cos 2theta -2 cos theta-1)d theta`

=`[3 theta +2 sin 2 theta - 2 sin theta]_(-pi/3)^(pi/3)`

`=[3 pi/3 +2 sin ((2pi)/3) - 2 sin (pi/3)]-[-3pi/3-2sin ((2pi)/3) +2 sin (pi/3)]`

`= 2pi`