$$\underset{2}{\overset{9}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(4\; -\; 2)\; \cdot \; <mn>4</mn>}}{2}$ = $\frac{8}{2}$ = 4.

I₃ = $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₃ = $\frac{\mathrm{(7\; -\; 4)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>3</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-9}}{2}$ = -4.5.

I₄ = $$\underset{7}{\overset{9}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₄ = (9 - 7) ⋅ ( - 3 ) = 2 ⋅ ( - 3 ) = -6.

Somit gilt:

$$\underset{2}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ + I₄ = $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ + $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ + $$\underset{7}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = 4 -4.5 -6 = -6.5

= ${\left[-\frac{5}{3}{x}^3+2x\right]}_{-2}^0$

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

= $-\frac{5}{3}\cdot 0+0-\left(-\frac{5}{3}\cdot \left(-8\right)-4\right)$

= $0+0-\left(\frac{40}{3}-4\right)$

= $0-\left(\frac{40}{3}-\frac{12}{3}\right)$

= $-1·\frac{28}{3}$

= $-\frac{28}{3}$

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

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

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

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

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

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

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

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

= $4e^{-\frac{1}{2}\pi }-4e^{-\frac{3}{2}\pi }-\frac{8}{3}$

= ${\left[-\frac{1}{6}{\left(-2x+2\right)}^3\right]}_2^4$

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

= $-\frac{1}{6}\cdot {\left(-8+2\right)}^3+\frac{1}{6}\cdot {\left(-4+2\right)}^3$

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

= $-\frac{1}{6}\cdot \left(-216\right)+\frac{1}{6}\cdot \left(-8\right)$

= $36-\frac{4}{3}$

= $\frac{108}{3}-\frac{4}{3}$

= $\frac{104}{3}$

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

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

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

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

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

= ${\left[\frac{1}{3}{\left(-3x+7\right)}^3\right]}_2^4$

$=$ $\frac{1}{3}\cdot {\left(-3\cdot 4+7\right)}^3-\frac{1}{3}\cdot {\left(-3\cdot 2+7\right)}^3$

= $\frac{1}{3}\cdot {\left(-12+7\right)}^3-\frac{1}{3}\cdot {\left(-6+7\right)}^3$

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

= $\frac{1}{3}\cdot \left(-125\right)-\frac{1}{3}\cdot 1$

= $-\frac{125}{3}-\frac{1}{3}$

= $-42$