# How do you find the area of r=1+cos(theta)?

#### Risposta:

(3pi)/2 unità areali.

#### Spiegazione:

If the pole r = 0 is not outside the region, the area is given by

(1/2) int r^2 d theta, with appropriate limits.

The given curve is a closed curve called cardioid.

It passes through the pole r = 0 and is symmetrical about the initial

linea theta = 0.

As r = f(cos theta), r is periodic with period 2pi.

And so the area enclosed by the cardioid is

(1/2) int r^2 d theta, Oltre theta in [0, 2pi].

(1/2)(2) int (1+cos theta)^2 d theta, theta in [0, pi], using symmetry about theta=0

=int (1+cos theta)^2 d theta, theta in [0, pi]

=int (1+2 cos theta + cos^2theta) d theta, theta in [0, pi]

=int (1+2 cos theta + (1+cos 2theta)/2) d theta, theta in [0, pi]

=[3/2theta+2sin theta]+(1/2)(1/2)sin 2theta],

between limits 0 and pi

=(3pi)/2+0+0