$$\underset{2}{\overset{9}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(4\; -\; 2)\; \cdot \; <mn>2</mn>}}{2}$ = $\frac{4}{2}$ = 2.

I₃ = $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₃ = (7 - 4) ⋅ 2 = 3 ⋅ 2 = 6.

I₄ = $$\underset{7}{\overset{9}{\int }}\mathrm{f(x)}dx$$ : Trapezfläche I₄ = (9 - 7) ⋅ $\frac{\mathrm{2\; +\; <mn>3</mn>}}{2}$ = 2 ⋅ 2.5 = 5.

Somit gilt:

$$\underset{2}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ + I₄ = $$\underset{2}{\overset{4}{\int }}\mathrm{f(x)}dx$$ + $$\underset{4}{\overset{7}{\int }}\mathrm{f(x)}dx$$ + $$\underset{7}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = 2 +6 +5 = 13

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

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

= $8+8-\left(1+4\right)$

= $8+8-1·1-1·4$

= $8+8-1-4$

= $11$

= ${\left[4x^{-2}+\frac{5}{9}{e}^{3x}\right]}_2^3$

= ${\left[\frac{4}{x^2}+\frac{5}{9}{e}^{3x}\right]}_2^3$

$=$ $\frac{4}{3^2}+\frac{5}{9}{e}^{3\cdot 3}-\left(\frac{4}{2^2}+\frac{5}{9}{e}^{3\cdot 2}\right)$

= $4\cdot \left(\frac{1}{9}\right)+\frac{5}{9}{e}^9-\left(4\cdot \left(\frac{1}{4}\right)+\frac{5}{9}{e}^6\right)$

= $\frac{4}{9}+\frac{5}{9}{e}^9-\left(1+\frac{5}{9}{e}^6\right)$

= $\frac{5}{9}{e}^9+\frac{4}{9}-\left(\frac{5}{9}{e}^6+1\right)$

= $\frac{5}{9}{e}^9+\frac{4}{9}-\frac{5}{9}{e}^6-1·1$

= $\frac{5}{9}{e}^9+\frac{4}{9}-\frac{5}{9}{e}^6-1$

= $\frac{5}{9}{e}^9-\frac{5}{9}{e}^6+\frac{4}{9}-1$

= $\frac{5}{9}{e}^9-\frac{5}{9}{e}^6-\frac{5}{9}$

= ${\left[\frac{1}{9}{\left(-3x+5\right)}^3+x^2\right]}_2^4$

$=$ $\frac{1}{9}\cdot {\left(-3\cdot 4+5\right)}^3+4^2-\left(\frac{1}{9}\cdot {\left(-3\cdot 2+5\right)}^3+2^2\right)$

= $\frac{1}{9}\cdot {\left(-12+5\right)}^3+16-\left(\frac{1}{9}\cdot {\left(-6+5\right)}^3+4\right)$

= $\frac{1}{9}\cdot {\left(-7\right)}^3+16-\left(\frac{1}{9}\cdot {\left(-1\right)}^3+4\right)$

= $\frac{1}{9}\cdot \left(-343\right)+16-\left(\frac{1}{9}\cdot \left(-1\right)+4\right)$

= $-\frac{343}{9}+16-\left(-\frac{1}{9}+4\right)$

= $-\frac{343}{9}+\frac{144}{9}-\left(-\frac{1}{9}+\frac{36}{9}\right)$

= $-\frac{199}{9}-1·\frac{35}{9}$

= $-\frac{199}{9}-\frac{35}{9}$

= $-26$

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

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

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

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

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

= $-\frac{3}{8\pi }-\left(\frac{1}{2}-\frac{3}{2\pi }\right)$

= $-1·\frac{1}{2}-1·\left(-\frac{3}{2\pi }\right)-\frac{3}{8\pi }$

= $-\frac{1}{2}+\frac{3}{2\pi }-\frac{3}{8\pi }$

= $-\frac{1}{2}+\frac{9}{8\pi }$

= ${\left[e^{-3x+7}\right]}_0^2$

$=$ $e^{-3\cdot 2+7}-e^{-3\cdot 0+7}$

= $e^{-6+7}-e^{0+7}$

= $e-e^7$

= $e-e^7$