$$\underset{2}{\overset{5}{\int }}\mathrm{f(x)}dx$$ 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:

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:

$$\underset{2}{\overset{5}{\int }}\mathrm{f(x)}dx$$ = I₂ = $$\underset{2}{\overset{5}{\int }}\mathrm{f(x)}dx$$ = -6 = -6

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

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

= $-1+5-\left(-1-5\right)$

= $-1+5-1·\left(-1\right)-1·\left(-5\right)$

= $-1+5+1+5$

= $10$

= ${\left[-\frac{7}{2}\cdot \sin \left(x\right)-2e^{-x}\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

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

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

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

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

= $-2e^{-\frac{3}{2}\pi }+\frac{7}{2}+2e^{-\frac{1}{2}\pi }-1·\left(-\frac{7}{2}\right)$

= $-2e^{-\frac{3}{2}\pi }+\frac{7}{2}+2e^{-\frac{1}{2}\pi }+\frac{7}{2}$

= $2e^{-\frac{1}{2}\pi }-2e^{-\frac{3}{2}\pi }+\frac{7}{2}+\frac{7}{2}$

= $2e^{-\frac{1}{2}\pi }-2e^{-\frac{3}{2}\pi }+7$

= ${\left[-\frac{3}{2}{e}^{2x-4}\right]}_0^2$

$=$ $-\frac{3}{2}{e}^{2\cdot 2-4}+\frac{3}{2}{e}^{2\cdot 0-4}$

= $-\frac{3}{2}{e}^{4-4}+\frac{3}{2}{e}^{0-4}$

= $-\frac{3}{2}{e}^0+\frac{3}{2}{e}^{-4}$

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

= ${\left[-\frac{1}{6}{x}^6+\frac{4}{3}\cdot \cos \left(x\right)\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

$=$ $-\frac{1}{6}\cdot {\left(\frac{3}{2}\pi \right)}^6+\frac{4}{3}\cdot \cos \left(\frac{3}{2}\pi \right)-\left(-\frac{1}{6}\cdot {\left(\frac{1}{2}\pi \right)}^6+\frac{4}{3}\cdot \cos \left(\frac{1}{2}\pi \right)\right)$

= $-\frac{1}{6}\cdot {\left(\frac{3}{2}\pi \right)}^6+\frac{4}{3}\cdot 0-\left(-\frac{1}{6}\cdot {\left(\frac{1}{2}\pi \right)}^6+\frac{4}{3}\cdot 0\right)$

= $-\frac{1}{6}\cdot {\left(\frac{3}{2}\pi \right)}^6+0-\left(-\frac{1}{6}\cdot {\left(\frac{1}{2}\pi \right)}^6+0\right)$

= $-\frac{243}{128}\cdot {\pi }^6+\frac{1}{384}\cdot {\pi }^6$

= $-\frac{729}{384}\cdot {\pi }^6+\frac{1}{384}\cdot {\pi }^6$

= $-\frac{91}{48}\cdot {\pi }^6$

= ${\left[\frac{1}{2}{e}^{2x-4}\right]}_0^1$

$=$ $\frac{1}{2}{e}^{2\cdot 1-4}-\frac{1}{2}{e}^{2\cdot 0-4}$

= $\frac{1}{2}{e}^{2-4}-\frac{1}{2}{e}^{0-4}$

= $\frac{1}{2}{e}^{-2}-\frac{1}{2}{e}^{-4}$