$$\underset{3}{\overset{9}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{7}{\overset{9}{\int }}\mathrm{f(x)}dx$$ : Trapezfläche I₄ = (9 - 7) ⋅ $\frac{\mathrm{1\; +\; <mn>3</mn>}}{2}$ = 2 ⋅ 2 = 4.

Somit gilt:

$$\underset{3}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ + I₄ = $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ + $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ + $$\underset{7}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = 1 +2 +4 = 7

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

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

= $-\frac{4}{3}\cdot 8+6-\left(-\frac{4}{3}\cdot \left(-1\right)-3\right)$

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

= $-\frac{32}{3}+\frac{18}{3}-\left(\frac{4}{3}-\frac{9}{3}\right)$

= $-\frac{14}{3}-1·\left(-\frac{5}{3}\right)$

= $-\frac{14}{3}+\frac{5}{3}$

= $-3$

= ${\left[7\cdot \cos \left(x\right)-\frac{1}{3}{x}^3\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

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

= $7\cdot 0-\frac{1}{3}\cdot {\left(\frac{3}{2}\pi \right)}^3-\left(7\cdot 0-\frac{1}{3}\cdot {\left(\frac{1}{2}\pi \right)}^3\right)$

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

= $-\frac{9}{8}\cdot {\pi }^3+\frac{1}{24}\cdot {\pi }^3$

= $-\frac{27}{24}\cdot {\pi }^3+\frac{1}{24}\cdot {\pi }^3$

= $-\frac{13}{12}\cdot {\pi }^3$

= ${\left[\frac{3}{2}{e}^{2x-4}\right]}_0^2$

$=$ $\frac{3}{2}{e}^{2\cdot 2-4}-\frac{3}{2}{e}^{2\cdot 0-4}$

= $\frac{3}{2}{e}^{4-4}-\frac{3}{2}{e}^{0-4}$

= $\frac{3}{2}{e}^0-\frac{3}{2}{e}^{-4}$

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

= ${\left[-6x^{\frac{3}{2}}+\frac{4}{3}{x}^3\right]}_1^{16}$

= ${\left[-6{\left(\sqrt{x}\right)}^3+\frac{4}{3}{x}^3\right]}_1^{16}$

$=$ $-6{\left(\sqrt{16}\right)}^3+\frac{4}{3}\cdot {16}^3-\left(-6{\left(\sqrt{1}\right)}^3+\frac{4}{3}\cdot 1^3\right)$

= $-6\cdot 4^3+\frac{4}{3}\cdot 4096-\left(-6\cdot 1^3+\frac{4}{3}\cdot 1\right)$

= $-6\cdot 64+\frac{16384}{3}-\left(-6\cdot 1+\frac{4}{3}\right)$

= $-384+\frac{16384}{3}-\left(-6+\frac{4}{3}\right)$

= $-\frac{1152}{3}+\frac{16384}{3}-\left(-\frac{18}{3}+\frac{4}{3}\right)$

= $\frac{15232}{3}-1·\left(-\frac{14}{3}\right)$

= $\frac{15232}{3}+\frac{14}{3}$

= $5082$

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

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

= $-e^{-9+3}+e^{-6+3}$

= $-e^{-6}+e^{-3}$