Find the area enclose by one loop of the curve
r= 2 Cos[x] - Sec[x]
We need the limits of integration first. since the loop goes through the origin, we need to find the values of x for which the curve passes through the origin, so set r = 0
we have,
looking at the graph, we see we want the region between and
we want to go anticlockwise from to to get the desired area, but we must go from a smaller to a larger angle. so rewrite as and we are in business
So our desired area is given by