$$\underset{3}{\overset{10}{\int }}\mathrm{f(x)}dx$$ 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 = $\frac{\mathrm{(6\; -\; 3)\; \cdot \; <mn>4</mn>}}{2}$ = $\frac{\mathrm{12}}{2}$ = 6.

I₃ = $$\underset{6}{\overset{8}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₃ = (8 - 6) ⋅ 4 = 2 ⋅ 4 = 8.

I₄ = $$\underset{8}{\overset{10}{\int }}\mathrm{f(x)}dx$$ : Trapezfläche I₄ = (10 - 8) ⋅ $\frac{\mathrm{4\; +\; <mn>3</mn>}}{2}$ = 2 ⋅ 3.5 = 7.

Somit gilt:

$$\underset{3}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ + I₄ = $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ + $$\underset{6}{\overset{8}{\int }}\mathrm{f(x)}dx$$ + $$\underset{8}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = 6 +8 +7 = 21

= ${\left[\frac{4}{3}{x}^3+3x\right]}_{-3}^0$

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

= $\frac{4}{3}\cdot 0+0-\left(\frac{4}{3}\cdot \left(-27\right)-9\right)$

= $0+0-\left(-36-9\right)$

= $0-1·\left(-45\right)$

= $45$

= ${\left[\frac{2}{3}{e}^{3x}-2x^{\frac{3}{2}}\right]}_4^9$

= ${\left[\frac{2}{3}{e}^{3x}-2{\left(\sqrt{x}\right)}^3\right]}_4^9$

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

= $\frac{2}{3}{e}^{27}-2\cdot 3^3-\left(\frac{2}{3}{e}^{12}-2\cdot 2^3\right)$

= $\frac{2}{3}{e}^{27}-2\cdot 27-\left(\frac{2}{3}{e}^{12}-2\cdot 8\right)$

= $\frac{2}{3}{e}^{27}-54-\left(\frac{2}{3}{e}^{12}-16\right)$

= $\frac{2}{3}{e}^{27}-54-\frac{2}{3}{e}^{12}-1·\left(-16\right)$

= $\frac{2}{3}{e}^{27}-54-\frac{2}{3}{e}^{12}+16$

= $\frac{2}{3}{e}^{27}-\frac{2}{3}{e}^{12}-54+16$

= $\frac{2}{3}{e}^{27}-\frac{2}{3}{e}^{12}-38$

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

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

= $-\frac{1}{2}\cdot {\left(-2\right)}^4-25-\left(-\frac{1}{2}\cdot 1^4-4\right)$

= $-\frac{1}{2}\cdot 16-25-\left(-\frac{1}{2}\cdot 1-4\right)$

= $-8-25-\left(-\frac{1}{2}-4\right)$

= $-33-\left(-\frac{1}{2}-\frac{8}{2}\right)$

= $-33-1·\left(-\frac{9}{2}\right)$

= $-33+\frac{9}{2}$

= $-\frac{66}{2}+\frac{9}{2}$

= $-33+\frac{9}{2}$

= $-\frac{66}{2}+\frac{9}{2}$

= $-\frac{57}{2}$

= ${\left[4\cdot \sin \left(x\right)+\frac{9}{4}{e}^{2x}\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

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

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

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

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

= $\frac{9}{4}{e}^{3\pi }-4-\frac{9}{4}{e}^{\pi }-1·4$

= $\frac{9}{4}{e}^{3\pi }-4-\frac{9}{4}{e}^{\pi }-4$

= $\frac{9}{4}{e}^{3\pi }-\frac{9}{4}{e}^{\pi }-4-4$

= $\frac{9}{4}{e}^{3\pi }-\frac{9}{4}{e}^{\pi }-8$

= ${\left[\frac{1}{6}{\left(2x-5\right)}^3+\frac{5}{2}{x}^2\right]}_1^3$

$=$ $\frac{1}{6}\cdot {\left(2\cdot 3-5\right)}^3+\frac{5}{2}\cdot 3^2-\left(\frac{1}{6}\cdot {\left(2\cdot 1-5\right)}^3+\frac{5}{2}\cdot 1^2\right)$

= $\frac{1}{6}\cdot {\left(6-5\right)}^3+\frac{5}{2}\cdot 9-\left(\frac{1}{6}\cdot {\left(2-5\right)}^3+\frac{5}{2}\cdot 1\right)$

= $\frac{1}{6}\cdot 1^3+\frac{45}{2}-\left(\frac{1}{6}\cdot {\left(-3\right)}^3+\frac{5}{2}\right)$

= $\frac{1}{6}\cdot 1+\frac{45}{2}-\left(\frac{1}{6}\cdot \left(-27\right)+\frac{5}{2}\right)$

= $\frac{1}{6}+\frac{45}{2}-\left(-\frac{9}{2}+\frac{5}{2}\right)$

= $\frac{1}{6}+\frac{135}{6}-1·\left(-2\right)$

= $\frac{68}{3}+2$

= $\frac{68}{3}+\frac{6}{3}$

= $\frac{68}{3}+2$

= $\frac{68}{3}+\frac{6}{3}$

= $\frac{74}{3}$