$$\underset{0}{\overset{9}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{0}{\overset{3}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₁ = $\frac{\mathrm{(3\; -\; 0)\; \cdot \; <mn>4</mn>}}{2}$ = $\frac{\mathrm{12}}{2}$ = 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 = $$\underset{6}{\overset{9}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₃ = (9 - 6) ⋅ ( - 2 ) = 3 ⋅ ( - 2 ) = -6.

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 -6 = -3

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

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

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

= $64+24-\left(1+\frac{3}{2}\right)$

= $88-\left(\frac{2}{2}+\frac{3}{2}\right)$

= $88-1·\frac{5}{2}$

= $88-\frac{5}{2}$

= $\frac{176}{2}-\frac{5}{2}$

= $\frac{171}{2}$

= ${\left[-\frac{3}{2}{x}^6-\frac{7}{4}{e}^{-2x}\right]}_0^3$

$=$ $-\frac{3}{2}\cdot 3^6-\frac{7}{4}{e}^{-2\cdot 3}-\left(-\frac{3}{2}\cdot 0^6-\frac{7}{4}{e}^{-2\cdot 0}\right)$

= $-\frac{3}{2}\cdot 729-\frac{7}{4}{e}^{-6}-\left(-\frac{3}{2}\cdot 0-\frac{7}{4}{e}^0\right)$

= $-\frac{2187}{2}-\frac{7}{4}{e}^{-6}-\left(0-\frac{7}{4}\right)$

= $-\frac{7}{4}{e}^{-6}-\frac{2187}{2}-\left(0-\frac{7}{4}\right)$

= $-\frac{7}{4}{e}^{-6}-\frac{2187}{2}+\frac{7}{4}$

= $-\frac{7}{4}{e}^{-6}-\frac{4367}{4}$

= ${\left[-\frac{2}{3}{\left(3x-4\right)}^{\frac{3}{2}}\right]}_{\frac{8}{3}}^{\frac{20}{3}}$

= ${\left[-\frac{2}{3}{\left(\sqrt{3x-4}\right)}^3\right]}_{\frac{8}{3}}^{\frac{20}{3}}$

$=$ $-\frac{2}{3}{\left(\sqrt{3\cdot \left(\frac{20}{3}\right)-4}\right)}^3+\frac{2}{3}{\left(\sqrt{3\cdot \left(\frac{8}{3}\right)-4}\right)}^3$

= $-\frac{2}{3}{\left(\sqrt{20-4}\right)}^3+\frac{2}{3}{\left(\sqrt{8-4}\right)}^3$

= $-\frac{2}{3}{\left(\sqrt{16}\right)}^3+\frac{2}{3}{\left(\sqrt{4}\right)}^3$

= $-\frac{2}{3}\cdot 4^3+\frac{2}{3}\cdot 2^3$

= $-\frac{2}{3}\cdot 64+\frac{2}{3}\cdot 8$

= $-\frac{128}{3}+\frac{16}{3}$

= $-\frac{112}{3}$

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

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

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

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

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

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

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

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

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

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

= $2+0$

= $2$