$$\underset{0}{\overset{7}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{0}{\overset{2}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₁ = $\frac{\mathrm{(2\; -\; 0)\; \cdot \; <mn>2</mn>}}{2}$ = $\frac{4}{2}$ = 2.

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

I₃ = $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₃ = (7 - 4) ⋅ ( - 2 ) = 3 ⋅ ( - 2 ) = -6.

Somit gilt:

$$\underset{0}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = I₁ + I₂ + I₃ = $$\underset{0}{\overset{2}{\int }}\mathrm{f(x)}dx$$ + $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ + $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = 2 -2 -6 = -6

= ${\left[-\frac{5}{2}{x}^2-x\right]}_0^1$

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

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

= $-\frac{5}{2}-1-\left(0+0\right)$

= $-\frac{5}{2}-\frac{2}{2}+0$

= $-\frac{7}{2}$

= ${\left[\frac{1}{15}{x}^5-2x\right]}_1^3$

$=$ $\frac{1}{15}\cdot 3^5-2\cdot 3-\left(\frac{1}{15}\cdot 1^5-2\cdot 1\right)$

= $\frac{1}{15}\cdot 243-6-\left(\frac{1}{15}\cdot 1-2\right)$

= $\frac{81}{5}-6-\left(\frac{1}{15}-2\right)$

= $\frac{81}{5}-\frac{30}{5}-\left(\frac{1}{15}-\frac{30}{15}\right)$

= $\frac{51}{5}-1·\left(-\frac{29}{15}\right)$

= $\frac{51}{5}+\frac{29}{15}$

= $\frac{153}{15}+\frac{29}{15}$

= $\frac{182}{15}$

= ${\left[\frac{1}{3}{\left(x-2\right)}^3+6x\right]}_0^2$

$=$ $\frac{1}{3}\cdot {\left(2-2\right)}^3+6\cdot 2-\left(\frac{1}{3}\cdot {\left(0-2\right)}^3+6\cdot 0\right)$

= $\frac{1}{3}\cdot 0^3+12-\left(\frac{1}{3}\cdot {\left(-2\right)}^3+0\right)$

= $\frac{1}{3}\cdot 0+12-\left(\frac{1}{3}\cdot \left(-8\right)+0\right)$

= $0+12-\left(-\frac{8}{3}+0\right)$

= $12-\left(-\frac{8}{3}+0\right)$

= $12+\frac{8}{3}$

= $\frac{36}{3}+\frac{8}{3}$

= $\frac{44}{3}$

= ${\left[\frac{9}{8}{e}^{-2x}+\frac{3}{2}\cdot \sin \left(x\right)\right]}_0^{\frac{1}{2}\pi }$

$=$ $\frac{9}{8}{e}^{-2\cdot \left(\frac{1}{2}\pi \right)}+\frac{3}{2}\cdot \sin \left(\frac{1}{2}\pi \right)-\left(\frac{9}{8}{e}^{-2\cdot \left(0\right)}+\frac{3}{2}\cdot \sin \left(0\right)\right)$

= $\frac{9}{8}{e}^{-2\cdot \left(\frac{1}{2}\pi \right)}+\frac{3}{2}\cdot 1-\left(\frac{9}{8}{e}^0+\frac{3}{2}\cdot 0\right)$

= $\frac{9}{8}{e}^{-2\cdot \left(\frac{1}{2}\pi \right)}+\frac{3}{2}-\left(\frac{9}{8}+0\right)$

= $\frac{9}{8}{e}^{-2\cdot \left(\frac{1}{2}\pi \right)}+\frac{3}{2}-\left(\frac{9}{8}+0\right)$

= $\frac{9}{8}{e}^{-2\cdot \left(\frac{1}{2}\pi \right)}+\frac{3}{2}-\frac{9}{8}$

= $\frac{9}{8}{e}^{-\pi }+\frac{3}{8}$

= ${\left[-\frac{2}{3}{e}^{-3x+6}\right]}_0^1$

$=$ $-\frac{2}{3}{e}^{-3\cdot 1+6}+\frac{2}{3}{e}^{-3\cdot 0+6}$

= $-\frac{2}{3}{e}^{-3+6}+\frac{2}{3}{e}^{0+6}$

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