Find the area of the region inside the graph?

In polar coordinates the area calculation is

integral((1/2)*r^2 d(theta)) over the appropriate interval.

In this case we have r = 5 - cos(theta), so one full cycle

is from theta = 0 to theta = 2pi.

(To save some time I will refer to theta as t from now on.)

So the area will be

integral(t=0 to 2pi)((1/2)*(5 - cos(t))^2 dt) =

(1/2)*integral(t=0 to 2pi)((25 - 10*cos(t) + cos^2(t)) dt), (A).

Now integral(cos^2(t) dt) =

integral((1/2)*(1 + cos(2t)) dt) =

(1/2)*t + (1/4)*sin(2t).

Thus integral (A) comes out to

(1/2)*[25*t - 10*cos(t) + (1/2)*t + (1/4)*sin(2t)],

which evaluated from t = 0 to t = 2pi is

(1/2)*[25*2pi - 10*cos(2pi) + (1/2)*2pi + (1/4)*sin(4pi)] -

(1/2)*[0 - 10*cos(0) + 0 + (1/4)*sin(0)] =

(1/2)*[51*pi - 10] - (1/2)*(-10) = (51/2)*pi.