$$\underset{3}{\overset{8}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(6\; -\; 3)\; \cdot \; <mn>3</mn>}}{2}$ = $\frac{9}{2}$ = 4.5.

I₃ = $$\underset{6}{\overset{8}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₃ = $\frac{\mathrm{(8\; -\; 6)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>3</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-6}}{2}$ = -3.

Somit gilt:

$$\underset{3}{\overset{8}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ = $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ + $$\underset{6}{\overset{8}{\int }}\mathrm{f(x)}dx$$ = 4.5 -3 = 1.5

= ${\left[-\frac{4}{3}{x}^3+2x\right]}_{-1}^1$

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

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

= $-\frac{4}{3}+2-\left(\frac{4}{3}-2\right)$

= $-\frac{4}{3}+\frac{6}{3}-\left(\frac{4}{3}-\frac{6}{3}\right)$

= $\frac{2}{3}-1·\left(-\frac{2}{3}\right)$

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

= $\frac{4}{3}$

= ${\left[\frac{1}{3}{x}^{-3}+\frac{7}{8}{e}^{-2x}\right]}_2^5$

= ${\left[\frac{1}{3x^3}+\frac{7}{8}{e}^{-2x}\right]}_2^5$

$=$ $\frac{1}{3\cdot 5^3}+\frac{7}{8}{e}^{-2\cdot 5}-\left(\frac{1}{3\cdot 2^3}+\frac{7}{8}{e}^{-2\cdot 2}\right)$

= $\frac{1}{3}\cdot \left(\frac{1}{125}\right)+\frac{7}{8}{e}^{-10}-\left(\frac{1}{3}\cdot \left(\frac{1}{8}\right)+\frac{7}{8}{e}^{-4}\right)$

= $\frac{1}{375}+\frac{7}{8}{e}^{-10}-\left(\frac{1}{24}+\frac{7}{8}{e}^{-4}\right)$

= $\frac{7}{8}{e}^{-10}+\frac{1}{375}-\left(\frac{7}{8}{e}^{-4}+\frac{1}{24}\right)$

= $-\frac{7}{8}{e}^{-4}-1·\frac{1}{24}+\frac{7}{8}{e}^{-10}+\frac{1}{375}$

= $-\frac{7}{8}{e}^{-4}-\frac{1}{24}+\frac{7}{8}{e}^{-10}+\frac{1}{375}$

= $-\frac{7}{8}{e}^{-4}+\frac{7}{8}{e}^{-10}-\frac{1}{24}+\frac{1}{375}$

= $-\frac{7}{8}{e}^{-4}+\frac{7}{8}{e}^{-10}-\frac{39}{1000}$

= ${\left[-\frac{1}{3}{e}^{3x-5}\right]}_2^3$

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

= $-\frac{1}{3}{e}^{9-5}+\frac{1}{3}{e}^{6-5}$

= $-\frac{1}{3}{e}^4+\frac{1}{3}e$

= $-\frac{1}{3}{e}^4+\frac{1}{3}e$

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

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

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

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

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

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

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

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

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

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

= $0+0$

= 0