$$\underset{2}{\overset{7}{\int }}\mathrm{f(x)}dx$$ gibt des orientierten (also mit Vorzeichen behafteten) Flächeninhalt zwischen dem Graph der Funktion und der x-Achse an.

Um diesen aus dem Schaubild zu berechnen unterteilen wir die Fläche in Rechtecke, Parallelogrammen 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 = $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₃ = (7 - 4) ⋅ 3 = 3 ⋅ 3 = 9.

Somit gilt:

$$\underset{2}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ = $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ + $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = 3 +9 = 12

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

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

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

= $-6-6-\left(-\frac{3}{2}+3\right)$

= $-12-\left(-\frac{3}{2}+\frac{6}{2}\right)$

= $-12-1·\frac{3}{2}$

= $-12-\frac{3}{2}$

= $-\frac{24}{2}-\frac{3}{2}$

= $-\frac{27}{2}$

= ${\left[-\frac{4}{9}{x}^6-\frac{1}{12}{e}^{3x}\right]}_{-3}^{-2}$

$=$ $-\frac{4}{9}\cdot {\left(-2\right)}^6-\frac{1}{12}{e}^{3\cdot \left(-2\right)}-\left(-\frac{4}{9}\cdot {\left(-3\right)}^6-\frac{1}{12}{e}^{3\cdot \left(-3\right)}\right)$

= $-\frac{4}{9}\cdot 64-\frac{1}{12}{e}^{-6}-\left(-\frac{4}{9}\cdot 729-\frac{1}{12}{e}^{-9}\right)$

= $-\frac{256}{9}-\frac{1}{12}{e}^{-6}-\left(-324-\frac{1}{12}{e}^{-9}\right)$

= $-\frac{1}{12}{e}^{-6}-\frac{256}{9}-\left(-\frac{1}{12}{e}^{-9}-324\right)$

= $-\frac{1}{12}{e}^{-6}-\frac{256}{9}+\frac{1}{12}{e}^{-9}-1·\left(-324\right)$

= $-\frac{1}{12}{e}^{-6}-\frac{256}{9}+\frac{1}{12}{e}^{-9}+324$

= $-\frac{1}{12}{e}^{-6}+\frac{1}{12}{e}^{-9}-\frac{256}{9}+324$

= $-\frac{1}{12}{e}^{-6}+\frac{1}{12}{e}^{-9}+\frac{2660}{9}$

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

$=$ $2\cdot \sin (\pi -\pi )-2\cdot \sin (\frac{1}{2}\pi -\pi )$

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

= $2\cdot 0-2\cdot \left(-1\right)$

= $0+2$

= $2$

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

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

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

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

= $\frac{45}{8}\cdot {\pi }^3-\frac{5}{24}\cdot {\pi }^3$

= $\frac{135}{24}\cdot {\pi }^3-\frac{5}{24}\cdot {\pi }^3$

= $\frac{65}{12}\cdot {\pi }^3$

= ${\left[e^{3x-4}\right]}_0^3$

$=$ $e^{3\cdot 3-4}-e^{3\cdot 0-4}$

= $e^{9-4}-e^{0-4}$

= $e^5-e^{-4}$