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

I₃ = $$\underset{6}{\overset{8}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₃ = $\frac{\mathrm{(8\; -\; 6)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>4</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-8}}{2}$ = -4.

Somit gilt:

$$\underset{3}{\overset{8}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ = $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ + $$\underset{6}{\overset{8}{\int }}\mathrm{f(x)}dx$$ = 3 -4 = -1

= ${\left[-\frac{4}{3}{x}^3-\frac{1}{2}{x}^2\right]}_0^4$

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

= $-\frac{4}{3}\cdot 64-\frac{1}{2}\cdot 16-\left(-\frac{4}{3}\cdot 0-\frac{1}{2}\cdot 0\right)$

= $-\frac{256}{3}-8-\left(0+0\right)$

= $-\frac{256}{3}-\frac{24}{3}+0$

= $-\frac{280}{3}$

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

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

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

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

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

= $\frac{2}{3}{e}^{-3\pi }-\frac{2}{3}{e}^{-\frac{3}{2}\pi }-1·6$

= $\frac{2}{3}{e}^{-3\pi }-\frac{2}{3}{e}^{-\frac{3}{2}\pi }-6$

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

= ${\left[\frac{1}{3}\cdot \sin \left(3x-\frac{3}{2}\pi \right)\right]}_0^{\frac{3}{2}\pi }$

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

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

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

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

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

= $-\frac{1}{3}$

= ${\left[-4\cdot \sin \left(x\right)-\frac{9}{4}{x}^{-1}\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

= ${\left[-4\cdot \sin \left(x\right)-\frac{9}{4x}\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

$=$ $-4\cdot \sin \left(\frac{3}{2}\pi \right)-\frac{9}{4\cdot \frac{3}{2}\pi }-\left(-4\cdot \sin \left(\frac{1}{2}\pi \right)-\frac{9}{4\cdot \frac{1}{2}\pi }\right)$

= $-4\cdot \left(-1\right)-\frac{9}{4\cdot \frac{3}{2}\pi }-\left(-4\cdot 1-\frac{9}{4\cdot \frac{1}{2}\pi }\right)$

= $4-\frac{9}{4\cdot \frac{3}{2}\pi }-\left(-4-\frac{9}{4\cdot \frac{1}{2}\pi }\right)$

= $4-\frac{3}{2\pi }-\left(-4-\frac{9}{2\pi }\right)$

= $4-\frac{3}{2\pi }-1·\left(-4\right)-1·\left(-\frac{9}{2\pi }\right)$

= $4-\frac{3}{2\pi }+4+\frac{9}{2\pi }$

= $4+4-\frac{3}{2\pi }+\frac{9}{2\pi }$

= $8+\frac{3}{\pi }$

= ${\left[-{\left(x-1\right)}^3\right]}_2^4$

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

= $-3^3+1^3$

= $-27+1$

= $-26$