$$\underset{3}{\overset{10}{\int }}\mathrm{f(x)}dx$$ 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:

I₂ = $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(5\; -\; 3)\; \cdot \; <mn>1</mn>}}{2}$ = $\frac{2}{2}$ = 1.

I₃ = $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₃ = (7 - 5) ⋅ 1 = 2 ⋅ 1 = 2.

I₄ = $$\underset{7}{\overset{10}{\int }}\mathrm{f(x)}dx$$ : Trapezfläche I₄ = (10 - 7) ⋅ $\frac{\mathrm{1\; +\; <mn>2</mn>}}{2}$ = 3 ⋅ 1.5 = 4.5.

Somit gilt:

$$\underset{3}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ + I₄ = $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ + $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ + $$\underset{7}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = 1 +2 +4.5 = 7.5

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

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

= $\frac{4}{3}\cdot 8+2-\left(\frac{4}{3}\cdot 1+1\right)$

= $\frac{32}{3}+2-\left(\frac{4}{3}+1\right)$

= $\frac{32}{3}+\frac{6}{3}-\left(\frac{4}{3}+\frac{3}{3}\right)$

= $\frac{38}{3}-1·\frac{7}{3}$

= $\frac{38}{3}-\frac{7}{3}$

= $\frac{31}{3}$

= ${\left[\frac{3}{2}\cdot \sin \left(x\right)+\frac{5}{9}{e}^{-3x}\right]}_{\frac{1}{2}\pi }^{\pi }$

$=$ $\frac{3}{2}\cdot \sin \left(\pi \right)+\frac{5}{9}{e}^{-3\cdot \pi }-\left(\frac{3}{2}\cdot \sin \left(\frac{1}{2}\pi \right)+\frac{5}{9}{e}^{-3\cdot \left(\frac{1}{2}\pi \right)}\right)$

= $\frac{3}{2}\cdot 0+\frac{5}{9}{e}^{-3\cdot \pi }-\left(\frac{3}{2}\cdot 1+\frac{5}{9}{e}^{-3\cdot \left(\frac{1}{2}\pi \right)}\right)$

= $0+\frac{5}{9}{e}^{-3\cdot \pi }-\left(\frac{3}{2}+\frac{5}{9}{e}^{-3\cdot \left(\frac{1}{2}\pi \right)}\right)$

= $\frac{5}{9}{e}^{-3\pi }-\left(\frac{5}{9}{e}^{-\frac{3}{2}\pi }+\frac{3}{2}\right)$

= $\frac{5}{9}{e}^{-3\pi }-\frac{5}{9}{e}^{-\frac{3}{2}\pi }-1·\frac{3}{2}$

= $\frac{5}{9}{e}^{-3\pi }-\frac{5}{9}{e}^{-\frac{3}{2}\pi }-\frac{3}{2}$

= $-\frac{5}{9}{e}^{-\frac{3}{2}\pi }+\frac{5}{9}{e}^{-3\pi }-\frac{3}{2}$

= ${\left[-\frac{2}{3}{\left(2x-5\right)}^{\frac{3}{2}}\right]}_7^{\frac{21}{2}}$

= ${\left[-\frac{2}{3}{\left(\sqrt{2x-5}\right)}^3\right]}_7^{\frac{21}{2}}$

$=$ $-\frac{2}{3}{\left(\sqrt{2\cdot \left(\frac{21}{2}\right)-5}\right)}^3+\frac{2}{3}{\left(\sqrt{2\cdot 7-5}\right)}^3$

= $-\frac{2}{3}{\left(\sqrt{21-5}\right)}^3+\frac{2}{3}{\left(\sqrt{14-5}\right)}^3$

= $-\frac{2}{3}{\left(\sqrt{16}\right)}^3+\frac{2}{3}{\left(\sqrt{9}\right)}^3$

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

= $-\frac{2}{3}\cdot 64+\frac{2}{3}\cdot 27$

= $-\frac{128}{3}+18$

= $-\frac{128}{3}+\frac{54}{3}$

= $-\frac{74}{3}$

= ${\left[9\cdot \sin \left(x\right)+\frac{7}{2}\cdot \cos \left(x\right)\right]}_{\frac{1}{2}\pi }^{\pi }$

$=$ $9\cdot \sin \left(\pi \right)+\frac{7}{2}\cdot \cos \left(\pi \right)-\left(9\cdot \sin \left(\frac{1}{2}\pi \right)+\frac{7}{2}\cdot \cos \left(\frac{1}{2}\pi \right)\right)$

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

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

= $0-\frac{7}{2}-9$

= $-\frac{7}{2}-9$

= $-\frac{7}{2}-\frac{18}{2}$

= $-\frac{25}{2}$

= ${\left[3e^{x-2}\right]}_2^4$

$=$ $3e^{4-2}-3e^{2-2}$

= $3e^2-3e^0$

= $3e^2-3$