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

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

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

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

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

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

= $\frac{1}{2}-\frac{4}{2}+0$

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

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

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

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

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

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

= $\frac{1}{2}{e}^{\frac{9}{2}\pi }-3-\frac{1}{2}{e}^{\frac{3}{2}\pi }-1·3$

= $\frac{1}{2}{e}^{\frac{9}{2}\pi }-3-\frac{1}{2}{e}^{\frac{3}{2}\pi }-3$

= $\frac{1}{2}{e}^{\frac{9}{2}\pi }-\frac{1}{2}{e}^{\frac{3}{2}\pi }-3-3$

= $\frac{1}{2}{e}^{\frac{9}{2}\pi }-\frac{1}{2}{e}^{\frac{3}{2}\pi }-6$

= ${\left[{\left(x-2\right)}^{-2}\right]}_4^7$

= ${\left[\frac{1}{{\left(x-2\right)}^2}\right]}_4^7$

$=$ $\frac{1}{{\left(7-2\right)}^2}-\frac{1}{{\left(4-2\right)}^2}$

= $\frac{1}{5^2}-\frac{1}{2^2}$

= $\frac{1}{25}-\left(\frac{1}{4}\right)$

= $\frac{4}{100}-\frac{25}{100}$

= $-\frac{21}{100}$

= ${\left[-\frac{3}{2}{x}^{\frac{4}{3}}+5e^x\right]}_0^9$

= ${\left[-\frac{3}{2}{\left(\sqrt[3]{x}\right)}^4+5e^x\right]}_0^9$

$=$ $-\frac{3}{2}{\left(\sqrt[3]{9}\right)}^4+5e^9-\left(-\frac{3}{2}{\left(\sqrt[3]{0}\right)}^4+5e^0\right)$

= $-\frac{3}{2}\cdot 1^4+5e^9-\left(-\frac{3}{2}\cdot 0^4+5\right)$

= $-\frac{3}{2}\cdot 1+5e^9-\left(-\frac{3}{2}\cdot 0+5\right)$

= $-\frac{3}{2}+5e^9-\left(0+5\right)$

= $5e^9-\frac{3}{2}-5$

= $5e^9-\frac{13}{2}$

= ${\left[\frac{2}{3}{e}^{-3x+7}\right]}_1^3$

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

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

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