$$\underset{0}{\overset{9}{\int }}\mathrm{f(x)}dx$$ 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:

I₁ = $$\underset{0}{\overset{3}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₁ = (3 - 0) ⋅ ( - 2 ) = 3 ⋅ ( - 2 ) = -6.

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 = $$\underset{6}{\overset{9}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₃ = $\frac{\mathrm{(9\; -\; 6)\; \cdot \; <mn>2</mn>}}{2}$ = $\frac{6}{2}$ = 3.

Somit gilt:

$$\underset{0}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = I₁ + I₂ + I₃ = $$\underset{0}{\overset{3}{\int }}\mathrm{f(x)}dx$$ + $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ + $$\underset{6}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = -6 -3 +3 = -6

= ${\left[\frac{1}{2}{x}^2+5x\right]}_{-1}^3$

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

= $\frac{1}{2}\cdot 9+15-\left(\frac{1}{2}\cdot 1-5\right)$

= $\frac{9}{2}+15-\left(\frac{1}{2}-5\right)$

= $\frac{9}{2}+\frac{30}{2}-\left(\frac{1}{2}-\frac{10}{2}\right)$

= $\frac{39}{2}-1·\left(-\frac{9}{2}\right)$

= $\frac{39}{2}+\frac{9}{2}$

= $24$

= ${\left[5\cdot \cos \left(x\right)-\frac{3}{8}{e}^{2x}\right]}_0^{\frac{1}{2}\pi }$

$=$ $5\cdot \cos \left(\frac{1}{2}\pi \right)-\frac{3}{8}{e}^{2\cdot \left(\frac{1}{2}\pi \right)}-\left(5\cdot \cos \left(0\right)-\frac{3}{8}{e}^{2\cdot \left(0\right)}\right)$

= $5\cdot 0-\frac{3}{8}{e}^{2\cdot \left(\frac{1}{2}\pi \right)}-\left(5\cdot 1-\frac{3}{8}{e}^0\right)$

= $0-\frac{3}{8}{e}^{2\cdot \left(\frac{1}{2}\pi \right)}-\left(5-\frac{3}{8}\right)$

= $-\frac{3}{8}{e}^{\pi }-\left(\frac{40}{8}-\frac{3}{8}\right)$

= $-\frac{3}{8}{e}^{\pi }-1·\frac{37}{8}$

= $-\frac{3}{8}{e}^{\pi }-\frac{37}{8}$

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

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

= $\cos \left(\pi \right)-\cos \left(\frac{1}{2}\pi \right)$

= $-1-0$

= $-1$

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

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

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

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

= $-\frac{3}{2}{e}^{\pi }-\left(-\frac{2}{6}-\frac{9}{6}\right)$

= $-\frac{3}{2}{e}^{\pi }-1·\left(-\frac{11}{6}\right)$

= $-\frac{3}{2}{e}^{\pi }+\frac{11}{6}$

= ${\left[\frac{1}{2}{\left(-x+1\right)}^4\right]}_1^3$

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

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

= $\frac{1}{2}\cdot 16-\frac{1}{2}\cdot 0$

= $8+0$

= $8$