Set up a definite integral to represent the area of the comm

Solution

As usual, a sketch of these two curves is essential. (A good reference is given below). The hardest thing is finding where the two curves intersect and deciding which curve to use in the integrand. Solving 3 - 2sin(a) = 3 - 2cos(a) gives sin(a) = cos(a) or tan(a) = 1 so a = pi/4 and 5pi/4. This gives the angle(s) at which the two curves intersect. A sketch of the two curves (which are called limacons) shows that for pi/4 < a < 5pi/4 the first curve r = 3 -2sin(a) forms the boundary of the common interior whilst for 5pi/4 < a< pi/4 the other curve takes this role. So the definite integral representing the common interior is 1/2 {int_{a=pi/4}^{a=5pi/4} [3 - 2sin(a)]^2 da + int_{a=5pi/4)^{a=pi/4) [3 + 2cos(a)]^2 da}. where _{ } and ^{ } means lower and upper limits respectively.