$$\underset{2}{\overset{6}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(4\; -\; 2)\; \cdot \; <mn>3</mn>}}{2}$ = $\frac{6}{2}$ = 3.

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

Somit gilt:

$$\underset{2}{\overset{6}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ = $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ + $$\underset{4}{\overset{6}{\int }}\mathrm{f(x)}dx$$ = 3 -3 = 0

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

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

= $-\frac{1}{3}\cdot 1+\frac{3}{2}\cdot 1-\left(-\frac{1}{3}\cdot \left(-27\right)+\frac{3}{2}\cdot 9\right)$

= $-\frac{1}{3}+\frac{3}{2}-\left(9+\frac{27}{2}\right)$

= $-\frac{2}{6}+\frac{9}{6}-\left(\frac{18}{2}+\frac{27}{2}\right)$

= $\frac{7}{6}-1·\frac{45}{2}$

= $\frac{7}{6}-\frac{45}{2}$

= $\frac{7}{6}-\frac{135}{6}$

= $-\frac{64}{3}$

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

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

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

= $-\frac{5}{2}\cdot \left(\frac{1}{25}\right)-\frac{7}{3}\cdot 125-\left(-\frac{5}{2}\cdot 1-\frac{7}{3}\cdot 1\right)$

= $-\frac{1}{10}-\frac{875}{3}-\left(-\frac{5}{2}-\frac{7}{3}\right)$

= $-\frac{3}{30}-\frac{8750}{30}-\left(-\frac{15}{6}-\frac{14}{6}\right)$

= $-\frac{8753}{30}-1·\left(-\frac{29}{6}\right)$

= $-\frac{8753}{30}+\frac{29}{6}$

= $-\frac{8753}{30}+\frac{145}{30}$

= $-\frac{4304}{15}$

= ${\left[-\frac{2}{3}{e}^{3x-7}\right]}_2^4$

$=$ $-\frac{2}{3}{e}^{3\cdot 4-7}+\frac{2}{3}{e}^{3\cdot 2-7}$

= $-\frac{2}{3}{e}^{12-7}+\frac{2}{3}{e}^{6-7}$

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

= ${\left[\frac{2}{3}{x}^6-\frac{4}{3}\cdot \cos \left(x\right)\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

$=$ $\frac{2}{3}\cdot {\left(\frac{3}{2}\pi \right)}^6-\frac{4}{3}\cdot \cos \left(\frac{3}{2}\pi \right)-\left(\frac{2}{3}\cdot {\left(\frac{1}{2}\pi \right)}^6-\frac{4}{3}\cdot \cos \left(\frac{1}{2}\pi \right)\right)$

= $\frac{2}{3}\cdot {\left(\frac{3}{2}\pi \right)}^6-\frac{4}{3}\cdot 0-\left(\frac{2}{3}\cdot {\left(\frac{1}{2}\pi \right)}^6-\frac{4}{3}\cdot 0\right)$

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

= $\frac{243}{32}\cdot {\pi }^6-\frac{1}{96}\cdot {\pi }^6$

= $\frac{729}{96}\cdot {\pi }^6-\frac{1}{96}\cdot {\pi }^6$

= $\frac{91}{12}\cdot {\pi }^6$

= ${\left[e^{x-1}\right]}_2^4$

$=$ $e^{4-1}-e^{2-1}$

= $e^3-e$

= $e^3-e$