Příklad 4.46

[!example] Spočtěte obsah obrazce ohraničeného funkcemi $\cos{(\frac{\pi{x}}{2})}$ a $x^2-1$

[!info] $$\begin{aligned} f(x) &= \cos{(\frac{\pi{x}}{2})} \\ g(x) &= x^2-1 \end{aligned}$$

[!info] Vypočítání [[Integrační meze]] $f(x) = 0, x = 1, -1, 3, -3, ...$ $g(x) = 0, x = 1, -1$ - Integrál budeme počítat na intervalu $\left<-1, 1\right>$ - Monotonie funkce $f(x)$ ![[priklad_46_funkce_f.png]] - Monotonie funkce $g(x)$ ![[priklad_46_funkce_g.png]] $$ \begin{aligned} &= \int_{-1}^{1} f(x) - g(x)\;dx \ &= \int_{-1}^{1} \cos{\left(\frac{\pi{x}}{2}\right)}\;dx - \int_{-1}^{1}x^2\;dx+\int_{-1}^{1}1\;dx \ \end{aligned} $$

[!tip]+ Substituce

\[\large\begin{aligned} u &= \frac{\pi}{2}{x} \\ du &= \frac{\pi}{2} \\ \end{aligned}\]

\[ \begin{aligned} &= \int_{-1}^{1} \cos{\left(\frac{\pi{x}}{2}\right)}\;dx - \int_{-1}^{1}x^2\;dx+\int_{-1}^{1}1\;dx \\ &= \frac{2}{\pi}\int_{-1}^{1} \cos{\left(\frac{\pi{x}}{2}\right)}\;dx - \int_{-1}^{1}x^2\;dx+\int_{-1}^{1}1\;dx \\ &= \frac{2}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos{(u)}\;dx - \left[\frac{x^3}{3}\right]_{-1}^{1}+\left[x\right]_{-1}^{1} \\ &= \frac{2}{\pi}\left[\sin(u)\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} - \left[\frac{x^3}{3}\right]_{-1}^{1}+\left[x\right]_{-1}^{1} \\ &= \frac{2}{\pi}\left[1 + 1\right] - \frac{2}{3} + 2 \\ &= \frac{4}{\pi}-\frac{2}{3}+2 \\ &= \boxed{\frac{4}{\pi}+\frac{4}{3}} \end{aligned} \]