$$\underset{0}{\overset{10}{\int }}\mathrm{f(x)}dx$$ 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 = $\frac{\mathrm{(7\; -\; 4)\; \cdot \; <mn>3</mn>}}{2}$ = $\frac{9}{2}$ = 4.5.

I₄ = $$\underset{7}{\overset{10}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₄ = (10 - 7) ⋅ 3 = 3 ⋅ 3 = 9.

Somit gilt:

$$\underset{0}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = I₁ + I₂ + I₃ + I₄ = $$\underset{0}{\overset{2}{\int }}\mathrm{f(x)}dx$$ + $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ + $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ + $$\underset{7}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = -4 -2 +4.5 +9 = 7.5

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

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

= $\frac{2}{3}\cdot \left(-8\right)-\frac{5}{2}\cdot 4-\left(\frac{2}{3}\cdot \left(-27\right)-\frac{5}{2}\cdot 9\right)$

= $-\frac{16}{3}-10-\left(-18-\frac{45}{2}\right)$

= $-\frac{16}{3}-\frac{30}{3}-\left(-\frac{36}{2}-\frac{45}{2}\right)$

= $-\frac{46}{3}-1·\left(-\frac{81}{2}\right)$

= $-\frac{46}{3}+\frac{81}{2}$

= $-\frac{92}{6}+\frac{243}{6}$

= $\frac{151}{6}$

= ${\left[\frac{7}{2}\cdot \cos \left(x\right)+\frac{5}{4}{x}^{-1}\right]}_{\pi }^{\frac{3}{2}\pi }$

= ${\left[\frac{7}{2}\cdot \cos \left(x\right)+\frac{5}{4x}\right]}_{\pi }^{\frac{3}{2}\pi }$

$=$ $\frac{7}{2}\cdot \cos \left(\frac{3}{2}\pi \right)+\frac{5}{4\cdot \frac{3}{2}\pi }-\left(\frac{7}{2}\cdot \cos \left(\pi \right)+\frac{5}{4\pi }\right)$

= $\frac{7}{2}\cdot 0+\frac{5}{4\cdot \frac{3}{2}\pi }-\left(\frac{7}{2}\cdot \left(-1\right)+\frac{5}{4\pi }\right)$

= $0+\frac{5}{4\cdot \frac{3}{2}\pi }-\left(-\frac{7}{2}+\frac{5}{4\pi }\right)$

= $\frac{5}{6\pi }-\left(-\frac{7}{2}+\frac{5}{4\pi }\right)$

= $-1·\left(-\frac{7}{2}\right)-1·\frac{5}{4\pi }+\frac{5}{6\pi }$

= $\frac{7}{2}-\frac{5}{4\pi }+\frac{5}{6\pi }$

= $\frac{7}{2}-\frac{5}{12\pi }$

= ${\left[e^{3x-6}\right]}_2^3$

$=$ $e^{3\cdot 3-6}-e^{3\cdot 2-6}$

= $e^{9-6}-e^{6-6}$

= $e^3-e^0$

= $e^3-1$

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

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

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

= $-\frac{5}{3}-e^{-2}-\left(\frac{5}{3}-e^2\right)$

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

= $e^2-1·\frac{5}{3}-e^{-2}-\frac{5}{3}$

= $e^2-\frac{5}{3}-e^{-2}-\frac{5}{3}$

= $e^2-e^{-2}-\frac{5}{3}-\frac{5}{3}$

= $e^2-e^{-2}-\frac{10}{3}$

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

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

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

= $-3\cdot \left(-1\right)+3\cdot 0$

= $3+0$

= $3$