$$\underset{0}{\overset{10}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{0}{\overset{3}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₁ = $\frac{\mathrm{(3\; -\; 0)\; \cdot \; <mn>4</mn>}}{2}$ = $\frac{\mathrm{12}}{2}$ = 6.

I₂ = $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(5\; -\; 3)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>4</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-8}}{2}$ = -4.

I₃ = $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₃ = (7 - 5) ⋅ ( - 4 ) = 2 ⋅ ( - 4 ) = -8.

I₄ = $$\underset{7}{\overset{10}{\int }}\mathrm{f(x)}dx$$ : Trapezfläche I₄ = (10 - 7) ⋅ $\frac{\mathrm{-4\; +\; <mo>(</mo>\; <mo>-</mo>\; <mn>5</mn>\; <mo>)</mo>}}{2}$ = 3 ⋅ ( - 4.5 ) = -13.5.

Somit gilt:

$$\underset{0}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = I₁ + I₂ + I₃ + I₄ = $$\underset{0}{\overset{3}{\int }}\mathrm{f(x)}dx$$ + $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ + $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ + $$\underset{7}{\overset{10}{\int }}\mathrm{f(x)}dx$$ = 6 -4 -8 -13.5 = -19.5

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

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

= $8-\frac{3}{2}\cdot 4-\left(0-\frac{3}{2}\cdot 0\right)$

= $8-6-\left(0+0\right)$

= $2+0$

= $2$

= ${\left[-2x^{\frac{3}{2}}-\frac{1}{2}{e}^{2x}\right]}_1^{16}$

= ${\left[-2{\left(\sqrt{x}\right)}^3-\frac{1}{2}{e}^{2x}\right]}_1^{16}$

$=$ $-2{\left(\sqrt{16}\right)}^3-\frac{1}{2}{e}^{2\cdot 16}-\left(-2{\left(\sqrt{1}\right)}^3-\frac{1}{2}{e}^{2\cdot 1}\right)$

= $-2\cdot 4^3-\frac{1}{2}{e}^{32}-\left(-2\cdot 1^3-\frac{1}{2}{e}^2\right)$

= $-2\cdot 64-\frac{1}{2}{e}^{32}-\left(-2\cdot 1-\frac{1}{2}{e}^2\right)$

= $-128-\frac{1}{2}{e}^{32}-\left(-2-\frac{1}{2}{e}^2\right)$

= $-\frac{1}{2}{e}^{32}-128-\left(-\frac{1}{2}{e}^2-2\right)$

= $-\frac{1}{2}{e}^{32}-128+\frac{1}{2}{e}^2-1·\left(-2\right)$

= $-\frac{1}{2}{e}^{32}-128+\frac{1}{2}{e}^2+2$

= $-\frac{1}{2}{e}^{32}+\frac{1}{2}{e}^2-128+2$

= $-\frac{1}{2}{e}^{32}+\frac{1}{2}{e}^2-126$

= ${\left[\frac{1}{9}{\left(-3x+6\right)}^3\right]}_2^4$

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

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

= $\frac{1}{9}\cdot {\left(-6\right)}^3-\frac{1}{9}\cdot 0^3$

= $\frac{1}{9}\cdot \left(-216\right)-\frac{1}{9}\cdot 0$

= $-24+0$

= $-24$

= ${\left[\frac{1}{8}{x}^{-2}-\cos \left(x\right)\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

= ${\left[\frac{1}{8x^2}-\cos \left(x\right)\right]}_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }$

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

= $\frac{1}{8{\left(\frac{3}{2}\pi \right)}^2}-0-\left(\frac{1}{8{\left(\frac{1}{2}\pi \right)}^2}-0\right)$

= $\frac{1}{18{\pi }^2}-\frac{1}{2{\pi }^2}$

= $-\frac{4}{9{\pi }^2}$

= ${\left[-\frac{2}{9}{\left(3x-7\right)}^3\right]}_1^2$

$=$ $-\frac{2}{9}\cdot {\left(3\cdot 2-7\right)}^3+\frac{2}{9}\cdot {\left(3\cdot 1-7\right)}^3$

= $-\frac{2}{9}\cdot {\left(6-7\right)}^3+\frac{2}{9}\cdot {\left(3-7\right)}^3$

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

= $-\frac{2}{9}\cdot \left(-1\right)+\frac{2}{9}\cdot \left(-64\right)$

= $\frac{2}{9}-\frac{128}{9}$

= $-14$