Question

Question #75f31

Answer 1

`96 pi`

Explanation:

The graph of the polar curve would be as shown below:

The area enclosed by the curve would be twice the area enclosed by the curve lying over x-axis. The formula for the area enccloseed by a polar curve is given by the formula `int_alpha^beta 1/2 r^2 d theta`.
In the present case `alpha =0` which comes by solving `r=16=8(1+ cos theta)` and `beta =pi`, which is obtained by solving `r=0=8(1+ cos theta)`

Accordingly, the desired area would be `2int_0^pi 1/2 r^2 d theta`

=`int_0^pi 64(1+cos theta)^2 d theta`

=`64int_0^pi (1 +2 cos theta + cos^2 theta) d theta`

=`64int_0^pi (1+2 cos theta +1/2 (1 +cos 2theta)d theta`

=`64[3/2 theta + 2sin theta +1/4 sin 2theta]_0^pi`

=`96pi`