$$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(6\; -\; 3)\; \cdot \; <mn>3</mn>}}{2}$ = $\frac{9}{2}$ = 4.5.

Somit gilt:

$$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ = I₂ = $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ = 4.5 = 4.5

= ${\left[\frac{2}{3}{x}^3-\frac{3}{2}{x}^2\right]}_{-2}^0$

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

= $\frac{2}{3}\cdot 0-\frac{3}{2}\cdot 0-\left(\frac{2}{3}\cdot \left(-8\right)-\frac{3}{2}\cdot 4\right)$

= $0+0-\left(-\frac{16}{3}-6\right)$

= $0-\left(-\frac{16}{3}-\frac{18}{3}\right)$

= $-1·\left(-\frac{34}{3}\right)$

= $\frac{34}{3}$

= ${\left[-\frac{5}{3}{x}^3+\frac{7}{9}{x}^{-3}\right]}_2^3$

= ${\left[-\frac{5}{3}{x}^3+\frac{7}{9x^3}\right]}_2^3$

$=$ $-\frac{5}{3}\cdot 3^3+\frac{7}{9\cdot 3^3}-\left(-\frac{5}{3}\cdot 2^3+\frac{7}{9\cdot 2^3}\right)$

= $-\frac{5}{3}\cdot 27+\frac{7}{9}\cdot \left(\frac{1}{27}\right)-\left(-\frac{5}{3}\cdot 8+\frac{7}{9}\cdot \left(\frac{1}{8}\right)\right)$

= $-45+\frac{7}{243}-\left(-\frac{40}{3}+\frac{7}{72}\right)$

= $-\frac{10935}{243}+\frac{7}{243}-\left(-\frac{960}{72}+\frac{7}{72}\right)$

= $-\frac{10928}{243}-1·\left(-\frac{953}{72}\right)$

= $-\frac{10928}{243}+\frac{953}{72}$

= $-\frac{87424}{1944}+\frac{25731}{1944}$

= $-\frac{61693}{1944}$

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

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

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

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

= $-\frac{1}{4}\cdot 1+1-\left(-\frac{1}{4}\cdot 256+0\right)$

= $-\frac{1}{4}+1-\left(-64+0\right)$

= $-\frac{1}{4}+\frac{4}{4}+64$

= $\frac{3}{4}+64$

= $\frac{3}{4}+\frac{256}{4}$

= $\frac{3}{4}+64$

= $\frac{259}{4}$

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

$=$ $\frac{5}{2}\cdot \cos \left(\pi \right)-2\cdot \sin \left(\pi \right)-\left(\frac{5}{2}\cdot \cos \left(\frac{1}{2}\pi \right)-2\cdot \sin \left(\frac{1}{2}\pi \right)\right)$

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

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

= $-\frac{5}{2}+0+2$

= $-\frac{5}{2}+2$

= $-\frac{5}{2}+\frac{4}{2}$

= $-\frac{1}{2}$

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

$=$ $3e^{-3+3}-3e^{-1+3}$

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

= $3-3e^2$