$$\underset{0}{\overset{10}{\int }}\mathrm{f(x)}dx$$ 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:

I₁ = $$\underset{0}{\overset{3}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₁ = (3 - 0) ⋅ 3 = 3 ⋅ 3 = 9.

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.

I₃ = $$\underset{6}{\overset{8}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₃ = $\frac{\mathrm{(8\; -\; 6)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>2</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-4}}{2}$ = -2.

I₄ = $$\underset{8}{\overset{10}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₄ = (10 - 8) ⋅ ( - 2 ) = 2 ⋅ ( - 2 ) = -4.

Somit gilt:

$$\underset{0}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = I₁ + I₂ + I₃ + I₄ = $$\underset{0}{\overset{3}{\int }}\mathrm{f(x)}dx$$ + $$\underset{3}{\overset{6}{\int }}\mathrm{f(x)}dx$$ + $$\underset{6}{\overset{8}{\int }}\mathrm{f(x)}dx$$ + $$\underset{8}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = 9 +4.5 -2 -4 = 7.5

= ${\left[-x^2-x\right]}_0^1$

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

= $-1-1-\left(-0+0\right)$

= $-1-1$

= $-2$

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

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

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

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

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

= $\frac{1}{2}{e}^{\frac{3}{2}\pi }-4-\frac{1}{2}{e}^{\frac{1}{2}\pi }-1·4$

= $\frac{1}{2}{e}^{\frac{3}{2}\pi }-4-\frac{1}{2}{e}^{\frac{1}{2}\pi }-4$

= $\frac{1}{2}{e}^{\frac{3}{2}\pi }-\frac{1}{2}{e}^{\frac{1}{2}\pi }-4-4$

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

= ${\left[-\frac{1}{6}{\left(-3x+6\right)}^4\right]}_0^3$

$=$ $-\frac{1}{6}\cdot {\left(-3\cdot 3+6\right)}^4+\frac{1}{6}\cdot {\left(-3\cdot 0+6\right)}^4$

= $-\frac{1}{6}\cdot {\left(-9+6\right)}^4+\frac{1}{6}\cdot {\left(0+6\right)}^4$

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

= $-\frac{1}{6}\cdot 81+\frac{1}{6}\cdot 1296$

= $-\frac{27}{2}+216$

= $-\frac{27}{2}+\frac{432}{2}$

= $\frac{405}{2}$

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

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

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

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

= $-\frac{1}{3}{e}^{3\cdot \pi }-4-\left(-\frac{1}{3}+\frac{12}{3}\right)$

= $-\frac{1}{3}{e}^{3\cdot \pi }-4-1·\frac{11}{3}$

= $-\frac{1}{3}{e}^{3\cdot \pi }-4-\frac{11}{3}$

= $-\frac{1}{3}{e}^{3\pi }-\frac{23}{3}$

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

$=$ $-e^{-2\cdot 3+2}+e^{-2\cdot 1+2}$

= $-e^{-6+2}+e^{-2+2}$

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

= $-e^{-4}+1$