Answer
$$A = \frac{2}{\pi}+\frac{2}{3}$$
Work Step by Step
From the graph, we can see that $\cos (\pi x) \geq 4x^{2}-1$ on the interval $\big[-\frac{1}{2},\frac{1}{2}\big]$; therefore, the area between the curves is $$ A= \displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}} \cos (\pi x) - (4x^{2}-1)dx$$ $$= \displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}} \cos(\pi x)-4x^{2}+1 dx$$ $$=\left[\frac{\sin (\pi x)}{\pi}-\frac{4x^{3}}{3}+x \right]^{\frac{1}{2}}_{-\frac{1}{2}}$$ $$=2\bigg[\frac{1}{\pi}-\frac{1}{6}+\frac{1}{2}\bigg]$$ $$=\frac{2}{\pi}+\frac{2}{3}$$
