$$f(x) = 1.5  \cdot Sin (\frac{\pi}{5}x + \frac{\pi}{2} ) + 1.5$$




下方蓝色曲线因为和上方紫色曲线对称
所以 下方曲线 g(x)=-f(x) $$f(x) = 1.5  \cdot Sin (\frac{\pi}{5}x + \frac{\pi}{2} ) + 1.5$$
$$g(x) = -1.5  \cdot Sin (\frac{\pi}{5}x + \frac{\pi}{2} ) - 1.5$$
需要求得的面积
$$F = \int_{-5}^5 [f(x)-g(x)] \,dx =\int_{-5}^5 (1.5  \cdot Sin (\frac{\pi}{5}x + \frac{\pi}{2} ) + 1.5 - [-1.5  \cdot Sin (\frac{\pi}{5}x + \frac{\pi}{2} ) - 1.5])\,dx$$
$$\Rightarrow F = \int_{-5}^5 [3 \cdot Sin (\frac{\pi}{5}x + \frac{\pi}{2} ) + 3]\,dx$$
$$\Rightarrow F = \int_{-5}^5 3 \cdot Sin (\frac{\pi}{5}x + \frac{\pi}{2})\,dx + \int_{-5}^5 3\,dx$$
$$\Rightarrow F =  \left|3 \cdot -Cos (\frac{\pi}{5}x + \frac{\pi}{2})\frac{5}{\pi}\right|_{-5}^5 +  \left|3x\right|_{-5}^5$$
$$\Rightarrow F = [ 3 \cdot -Cos (\frac{\pi}{5}\cdot5 + \frac{\pi}{2})\frac{5}{\pi}]- [3 \cdot -Cos (\frac{\pi}{5}\cdot(-5) + \frac{\pi}{2})\frac{5}{\pi}]+  3\cdot5-3\cdot(-5)$$
$$\Rightarrow F = [- \frac{15}{\pi}\cdot Cos (\pi + \frac{\pi}{2})]- [- \frac{15}{\pi}\cdot Cos (-\pi + \frac{\pi}{2})]+  3\cdot5-3\cdot(-5)$$
$$\Rightarrow F = [- \frac{15}{\pi}\cdot Cos (\frac{3}{2}\pi)]- [- \frac{15}{\pi}\cdot Cos (-\frac{\pi}{2})]+  15+15$$
$$\Rightarrow F = [- \frac{15}{\pi}\cdot 0]- [- \frac{15}{\pi}\cdot 0]+  15+15$$
$$\Rightarrow F =0+0+15+15$$
计算结果，阴影部分面积为： $$\Rightarrow F =30$$