$$\underset{2}{\overset{10}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{2}{\overset{5}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(5\; -\; 2)\; \cdot \; <mn>1</mn>}}{2}$ = $\frac{3}{2}$ = 1.5.

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:

$$\underset{2}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ + I₄ = $$\underset{2}{\overset{5}{\int }}\mathrm{f(x)}dx$$ + $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ + $$\underset{7}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = 1.5 +2 +4.5 = 8

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

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

= $-\frac{1}{3}\cdot 27-15-\left(-\frac{1}{3}\cdot 1-5\right)$

= $-9-15-\left(-\frac{1}{3}-5\right)$

= $-24-\left(-\frac{1}{3}-\frac{15}{3}\right)$

= $-24-1·\left(-\frac{16}{3}\right)$

= $-24+\frac{16}{3}$

= $-\frac{72}{3}+\frac{16}{3}$

= $-\frac{56}{3}$

= ${\left[\cos \left(x\right)-\frac{1}{2}{x}^4\right]}_0^{\frac{3}{2}\pi }$

$=$ $\cos \left(\frac{3}{2}\pi \right)-\frac{1}{2}\cdot {\left(\frac{3}{2}\pi \right)}^4-\left(\cos \left(0\right)-\frac{1}{2}\cdot {\left(0\right)}^4\right)$

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

= $-\frac{81}{32}\cdot {\pi }^4-\left(1+0\right)$

= $-\frac{81}{32}\cdot {\pi }^4-1$

= ${\left[\frac{3}{2}{e}^{2x-1}\right]}_1^4$

$=$ $\frac{3}{2}{e}^{2\cdot 4-1}-\frac{3}{2}{e}^{2\cdot 1-1}$

= $\frac{3}{2}{e}^{8-1}-\frac{3}{2}{e}^{2-1}$

= $\frac{3}{2}{e}^7-\frac{3}{2}e$

= $\frac{3}{2}{e}^7-\frac{3}{2}e$

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

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

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

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

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

= $-\frac{32}{81{\pi }^3}-\left(-\frac{5}{4}-\frac{4}{3{\pi }^3}\right)$

= $-1·\left(-\frac{5}{4}\right)-1·\left(-\frac{4}{3{\pi }^3}\right)-\frac{32}{81{\pi }^3}$

= $\frac{5}{4}+\frac{4}{3{\pi }^3}-\frac{32}{81{\pi }^3}$

= $\frac{5}{4}+\frac{76}{81{\pi }^3}$

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

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

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

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

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