$$\underset{3}{\overset{9}{\int }}\mathrm{f(x)}dx$$ 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 = $\frac{\mathrm{(6\; -\; 3)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>4</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-12}}{2}$ = -6.

I₃ = $$\underset{6}{\overset{9}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₃ = $\frac{\mathrm{(9\; -\; 6)\; \cdot \; <mn>4</mn>}}{2}$ = $\frac{\mathrm{12}}{2}$ = 6.

Somit gilt:

$$\underset{3}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ = $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ + $$\underset{6}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = -6 +6 = 0

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

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

= $\frac{3}{2}\cdot 4-6-\left(\frac{3}{2}\cdot 9-9\right)$

= $6-6-\left(\frac{27}{2}-9\right)$

= $0-\left(\frac{27}{2}-\frac{18}{2}\right)$

= $0-1·\frac{9}{2}$

= $0-\frac{9}{2}$

= $0-\frac{9}{2}$

= $-\frac{9}{2}$

= ${\left[-\frac{9}{2}{e}^{-2x}-\frac{5}{2}{x}^{-2}\right]}_1^3$

= ${\left[-\frac{9}{2}{e}^{-2x}-\frac{5}{2x^2}\right]}_1^3$

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

= $-\frac{9}{2}{e}^{-6}-\frac{5}{2}\cdot \left(\frac{1}{9}\right)-\left(-\frac{9}{2}{e}^{-2}-\frac{5}{2}\cdot 1\right)$

= $-\frac{9}{2}{e}^{-6}-\frac{5}{18}-\left(-\frac{9}{2}{e}^{-2}-\frac{5}{2}\right)$

= $\frac{9}{2}{e}^{-2}-1·\left(-\frac{5}{2}\right)-\frac{9}{2}{e}^{-6}-\frac{5}{18}$

= $\frac{9}{2}{e}^{-2}+\frac{5}{2}-\frac{9}{2}{e}^{-6}-\frac{5}{18}$

= $\frac{9}{2}{e}^{-2}-\frac{9}{2}{e}^{-6}+\frac{5}{2}-\frac{5}{18}$

= $\frac{9}{2}{e}^{-2}-\frac{9}{2}{e}^{-6}+\frac{20}{9}$

= ${\left[\frac{3}{4}{\left(x-3\right)}^4\right]}_1^4$

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

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

= $\frac{3}{4}\cdot 1-\frac{3}{4}\cdot 16$

= $\frac{3}{4}-12$

= $\frac{3}{4}-\frac{48}{4}$

= $-\frac{45}{4}$

= ${\left[\frac{4}{3}{e}^{3x}+\cos \left(x\right)\right]}_0^{\frac{1}{2}\pi }$

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

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

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

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

= $\frac{4}{3}{e}^{\frac{3}{2}\pi }-1·\frac{7}{3}$

= $\frac{4}{3}{e}^{\frac{3}{2}\pi }-\frac{7}{3}$

= ${\left[-\frac{1}{3}{\left(-3x+7\right)}^{-1}\right]}_3^5$

= ${\left[-\frac{1}{3\left(-3x+7\right)}\right]}_3^5$

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

= $-\frac{1}{3\left(-15+7\right)}+\frac{1}{3\left(-9+7\right)}$

= $-\frac{1}{3\left(-8\right)}+\frac{1}{3\left(-2\right)}$

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

= $\frac{1}{24}-\frac{1}{6}$

= $\frac{1}{24}-\frac{4}{24}$

= $-\frac{1}{8}$