Answer
$$A = 18$$
Work Step by Step
$$\eqalign{
& \left( i \right){\text{From the graph we can see that the net area of the region is:}} \cr
& A = \int_{ - \pi /2}^{\pi /2} {6\cos x} dx - \int_{\pi /2}^\pi {6\cos x} dx \cr
& {\text{Integrating}} \cr
& A = 6\left[ {\sin x} \right]_{ - \pi /2}^{\pi /2} - 6\left[ {\sin x} \right]_{\pi /2}^\pi \cr
& {\text{Evaluating}} \cr
& A = 6\left[ {\sin \left( {\frac{\pi }{2}} \right) - \sin \left( { - \frac{\pi }{2}} \right)} \right] - 6\left[ {\sin \left( \pi \right) - \sin \left( {\frac{\pi }{2}} \right)} \right] \cr
& A = 6\left( 2 \right) - 6\left[ {\sin \left( \pi \right) - \sin \left( {\frac{\pi }{2}} \right)} \right] \cr
& A = 12 + 6 \cr
& A = 18 \cr} $$
