$$\underset{0}{\overset{5}{\int }}\mathrm{f(x)}dx$$ gibt den orientierten (also mit Vorzeichen behafteten) Flächeninhalt zwischen dem Graph der Funktion f und der x-Achse an.

Um diesen aus dem Schaubild zu berechnen unterteilen wir die Fläche in Rechtecke, Parallelogramme und ggf. Dreiecke:

I₁ = $$\underset{0}{\overset{2}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₁ = (2 - 0) ⋅ ( - 4 ) = 2 ⋅ ( - 4 ) = -8.

I₂ = $$\underset{2}{\overset{5}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(5\; -\; 2)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>4</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-12}}{2}$ = -6.

Somit gilt:

$$\underset{0}{\overset{5}{\int }}\mathrm{f(x)}dx$$ = I₁ + I₂ = $$\underset{0}{\overset{2}{\int }}\mathrm{f(x)}dx$$ + $$\underset{2}{\overset{5}{\int }}\mathrm{f(x)}dx$$ = -8 -6 = -14

= ${\left[-\frac{5}{3}{x}^3+\frac{5}{2}{x}^2\right]}_{-1}^1$

$=$ $-\frac{5}{3}\cdot 1^3+\frac{5}{2}\cdot 1^2-\left(-\frac{5}{3}\cdot {\left(-1\right)}^3+\frac{5}{2}\cdot {\left(-1\right)}^2\right)$

= $-\frac{5}{3}\cdot 1+\frac{5}{2}\cdot 1-\left(-\frac{5}{3}\cdot \left(-1\right)+\frac{5}{2}\cdot 1\right)$

= $-\frac{5}{3}+\frac{5}{2}-\left(\frac{5}{3}+\frac{5}{2}\right)$

= $-\frac{10}{6}+\frac{15}{6}-\left(\frac{10}{6}+\frac{15}{6}\right)$

= $\frac{5}{6}-1·\frac{25}{6}$

= $\frac{5}{6}-\frac{25}{6}$

= $-\frac{10}{3}$

= ${\left[-\frac{7}{2}\cdot \cos \left(x\right)-\frac{3}{4}{e}^x\right]}_0^{\frac{3}{2}\pi }$

$=$ $-\frac{7}{2}\cdot \cos \left(\frac{3}{2}\pi \right)-\frac{3}{4}{e}^{\frac{3}{2}\pi }-\left(-\frac{7}{2}\cdot \cos \left(0\right)-\frac{3}{4}{e}^0\right)$

= $-\frac{7}{2}\cdot 0-\frac{3}{4}{e}^{\frac{3}{2}\pi }-\left(-\frac{7}{2}\cdot 1-\frac{3}{4}\right)$

= $0-\frac{3}{4}{e}^{\frac{3}{2}\pi }-\left(-\frac{7}{2}-\frac{3}{4}\right)$

= $-\frac{3}{4}{e}^{\frac{3}{2}\pi }-\left(-\frac{14}{4}-\frac{3}{4}\right)$

= $-\frac{3}{4}{e}^{\frac{3}{2}\pi }-1·\left(-\frac{17}{4}\right)$

= $-\frac{3}{4}{e}^{\frac{3}{2}\pi }+\frac{17}{4}$

= ${\left[-\frac{2}{3}{e}^{3x-7}\right]}_2^3$

$=$ $-\frac{2}{3}{e}^{3\cdot 3-7}+\frac{2}{3}{e}^{3\cdot 2-7}$

= $-\frac{2}{3}{e}^{9-7}+\frac{2}{3}{e}^{6-7}$

= $-\frac{2}{3}{e}^2+\frac{2}{3}{e}^{-1}$

= ${\left[3\cdot \sin \left(x\right)+\frac{2}{5}{x}^5\right]}_{\frac{1}{2}\pi }^{\pi }$

$=$ $3\cdot \sin \left(\pi \right)+\frac{2}{5}\cdot {\pi }^5-\left(3\cdot \sin \left(\frac{1}{2}\pi \right)+\frac{2}{5}\cdot {\left(\frac{1}{2}\pi \right)}^5\right)$

= $3\cdot 0+\frac{2}{5}\cdot {\pi }^5-\left(3\cdot 1+\frac{2}{5}\cdot {\left(\frac{1}{2}\pi \right)}^5\right)$

= $0+\frac{2}{5}\cdot {\pi }^5-\left(3+\frac{2}{5}\cdot {\left(\frac{1}{2}\pi \right)}^5\right)$

= $\frac{2}{5}\cdot {\pi }^5-\left(3+\frac{1}{80}\cdot {\pi }^5\right)$

= $-1·3-1·\frac{1}{80}\cdot {\pi }^5+\frac{2}{5}\cdot {\pi }^5$

= $-3-\frac{1}{80}\cdot {\pi }^5+\frac{2}{5}\cdot {\pi }^5$

= $-3+\frac{31}{80}\cdot {\pi }^5$

= ${\left[\frac{2}{3}\cdot \cos \left(3x+\frac{3}{2}\pi \right)\right]}_0^{\frac{3}{2}\pi }$

$=$ $\frac{2}{3}\cdot \cos \left(3\cdot \left(\frac{3}{2}\pi \right)+\frac{3}{2}\pi \right)-\frac{2}{3}\cdot \cos \left(3\cdot \left(0\right)+\frac{3}{2}\pi \right)$

= $\frac{2}{3}\cdot \cos \left(6\pi \right)-\frac{2}{3}\cdot \cos \left(\frac{3}{2}\pi \right)$

= $\frac{2}{3}\cdot 1-\frac{2}{3}\cdot 0$

= $\frac{2}{3}+0$

= $\frac{2}{3}+0$

= $\frac{2}{3}$