$$\underset{0}{\overset{7}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{0}{\overset{2}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₁ = (2 - 0) ⋅ ( - 2 ) = 2 ⋅ ( - 2 ) = -4.

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 = $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₃ = $\frac{\mathrm{(7\; -\; 4)\; \cdot \; <mn>2</mn>}}{2}$ = $\frac{6}{2}$ = 3.

Somit gilt:

$$\underset{0}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = I₁ + I₂ + I₃ = $$\underset{0}{\overset{2}{\int }}\mathrm{f(x)}dx$$ + $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ + $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = -4 -2 +3 = -3

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

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

= $-\frac{5}{2}\cdot 0+0-\left(-\frac{5}{2}\cdot 4-4\right)$

= $0+0-\left(-10-4\right)$

= $0-1·\left(-14\right)$

= $14$

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

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

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

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

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

= $-\frac{4}{3}{e}^{\frac{9}{2}\pi }+2+\frac{4}{3}{e}^{\frac{3}{2}\pi }-1·\left(-2\right)$

= $-\frac{4}{3}{e}^{\frac{9}{2}\pi }+2+\frac{4}{3}{e}^{\frac{3}{2}\pi }+2$

= $-\frac{4}{3}{e}^{\frac{9}{2}\pi }+\frac{4}{3}{e}^{\frac{3}{2}\pi }+2+2$

= $-\frac{4}{3}{e}^{\frac{9}{2}\pi }+\frac{4}{3}{e}^{\frac{3}{2}\pi }+4$

= ${\left[-\frac{1}{3}{\left(-3x+7\right)}^3+5x\right]}_1^2$

$=$ $-\frac{1}{3}\cdot {\left(-3\cdot 2+7\right)}^3+5\cdot 2-\left(-\frac{1}{3}\cdot {\left(-3\cdot 1+7\right)}^3+5\cdot 1\right)$

= $-\frac{1}{3}\cdot {\left(-6+7\right)}^3+10-\left(-\frac{1}{3}\cdot {\left(-3+7\right)}^3+5\right)$

= $-\frac{1}{3}\cdot 1^3+10-\left(-\frac{1}{3}\cdot 4^3+5\right)$

= $-\frac{1}{3}\cdot 1+10-\left(-\frac{1}{3}\cdot 64+5\right)$

= $-\frac{1}{3}+10-\left(-\frac{64}{3}+5\right)$

= $-\frac{1}{3}+\frac{30}{3}-\left(-\frac{64}{3}+\frac{15}{3}\right)$

= $\frac{29}{3}-1·\left(-\frac{49}{3}\right)$

= $\frac{29}{3}+\frac{49}{3}$

= $26$

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

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

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

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

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

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

= $-4+\frac{8}{3}\cdot {\pi }^3$

= ${\left[-\frac{2}{9}{\left(-3x+7\right)}^3-\frac{1}{2}{x}^2\right]}_0^1$

$=$ $-\frac{2}{9}\cdot {\left(-3\cdot 1+7\right)}^3-\frac{1}{2}\cdot 1^2-\left(-\frac{2}{9}\cdot {\left(-3\cdot 0+7\right)}^3-\frac{1}{2}\cdot 0^2\right)$

= $-\frac{2}{9}\cdot {\left(-3+7\right)}^3-\frac{1}{2}\cdot 1-\left(-\frac{2}{9}\cdot {\left(0+7\right)}^3-\frac{1}{2}\cdot 0\right)$

= $-\frac{2}{9}\cdot 4^3-\frac{1}{2}-\left(-\frac{2}{9}\cdot 7^3+0\right)$

= $-\frac{2}{9}\cdot 64-\frac{1}{2}-\left(-\frac{2}{9}\cdot 343+0\right)$

= $-\frac{128}{9}-\frac{1}{2}-\left(-\frac{686}{9}+0\right)$

= $-\frac{256}{18}-\frac{9}{18}-\left(-\frac{686}{9}+0\right)$

= $-\frac{265}{18}+\frac{686}{9}$

= $-\frac{265}{18}+\frac{1372}{18}$

= $-\frac{265}{18}+\frac{686}{9}$

= $-\frac{265}{18}+\frac{1372}{18}$

= $\frac{123}{2}$