Answer:
area = 14/π +4/3 ≈ 5.78967
Step-by-step explanation:
You want a sketch and the value of the area enclosed by the curves ...
Area
The attached graph shows the curves intersect at x = ±1/2, so those are the limits of integration. The area is symmetrical about the y-axis, so we can just integrate over [0, 1/2] and double the result.
[tex]\displaystyle A=2\int_0^{0.5}{(7\cos{(\pi x)}-(8x^2-2))}\,dx=2\left[\dfrac{7}{\pi}\sin{(\pi x)}-\dfrac{8}{3}x^3+2x\right]_0^{0.5}\\\\\\A=\dfrac{14}{\pi}-\dfrac{2}{3}+2=\boxed{\dfrac{14}{\pi}+\dfrac{4}{3}\approx 5.78967}[/tex]
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