$$\underset{0}{\overset{9}{\int }}\mathrm{f(x)}dx$$ gibt des orientierten (also mit Vorzeichen behafteten) Flächeninhalt zwischen dem Graph der Funktion und der x-Achse an.

Um diesen aus dem Schaubild zu berechnen unterteilen wir die Fläche in Rechtecke, Parallelogrammen und ggf. Dreiecke:

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

I₂ = $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(5\; -\; 3)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>3</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-6}}{2}$ = -3.

I₃ = $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₃ = $\frac{\mathrm{(7\; -\; 5)\; \cdot \; <mn>3</mn>}}{2}$ = $\frac{6}{2}$ = 3.

I₄ = $$\underset{7}{\overset{9}{\int }}\mathrm{f(x)}dx$$ : Rechtecksfläche I₄ = (9 - 7) ⋅ 3 = 2 ⋅ 3 = 6.

Somit gilt:

$$\underset{0}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = I₁ + I₂ + I₃ + I₄ = $$\underset{0}{\overset{3}{\int }}\mathrm{f(x)}dx$$ + $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ + $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ + $$\underset{7}{\overset{9}{\int }}\mathrm{f(x)}dx$$ = -9 -3 +3 +6 = -3

= ${\left[\frac{1}{3}{x}^3-2x^2\right]}_1^4$

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

= $\frac{1}{3}\cdot 64-2\cdot 16-\left(\frac{1}{3}\cdot 1-2\cdot 1\right)$

= $\frac{64}{3}-32-\left(\frac{1}{3}-2\right)$

= $\frac{64}{3}-\frac{96}{3}-\left(\frac{1}{3}-\frac{6}{3}\right)$

= $-\frac{32}{3}-1·\left(-\frac{5}{3}\right)$

= $-\frac{32}{3}+\frac{5}{3}$

= $-9$

= ${\left[\frac{1}{3}{x}^{-3}-\frac{7}{5}{x}^5\right]}_2^5$

= ${\left[\frac{1}{3x^3}-\frac{7}{5}{x}^5\right]}_2^5$

$=$ $\frac{1}{3\cdot 5^3}-\frac{7}{5}\cdot 5^5-\left(\frac{1}{3\cdot 2^3}-\frac{7}{5}\cdot 2^5\right)$

= $\frac{1}{3}\cdot \left(\frac{1}{125}\right)-\frac{7}{5}\cdot 3125-\left(\frac{1}{3}\cdot \left(\frac{1}{8}\right)-\frac{7}{5}\cdot 32\right)$

= $\frac{1}{375}-4375-\left(\frac{1}{24}-\frac{224}{5}\right)$

= $\frac{1}{375}-\frac{1640625}{375}-\left(\frac{5}{120}-\frac{5376}{120}\right)$

= $-\frac{1640624}{375}-1·\left(-\frac{5371}{120}\right)$

= $-\frac{1640624}{375}+\frac{5371}{120}$

= $-\frac{13124992}{3000}+\frac{134275}{3000}$

= $-\frac{4330239}{1000}$

= ${\left[\frac{1}{2}{\left(x-3\right)}^4-3x\right]}_1^4$

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

= $\frac{1}{2}\cdot 1^4-12-\left(\frac{1}{2}\cdot {\left(-2\right)}^4-3\right)$

= $\frac{1}{2}\cdot 1-12-\left(\frac{1}{2}\cdot 16-3\right)$

= $\frac{1}{2}-12-\left(8-3\right)$

= $\mathrm{0,5}-12-1·5$

= $-\mathrm{11,5}-5$

= $-\mathrm{16,5}$

= ${\left[-4\cdot \cos \left(x\right)+\frac{7}{2}{x}^{-2}\right]}_{\pi }^{\frac{3}{2}\pi }$

= ${\left[-4\cdot \cos \left(x\right)+\frac{7}{2x^2}\right]}_{\pi }^{\frac{3}{2}\pi }$

$=$ $-4\cdot \cos \left(\frac{3}{2}\pi \right)+\frac{7}{2{\left(\frac{3}{2}\pi \right)}^2}-\left(-4\cdot \cos \left(\pi \right)+\frac{7}{2{\pi }^2}\right)$

= $-4\cdot 0+\frac{7}{2{\left(\frac{3}{2}\pi \right)}^2}-\left(-4\cdot \left(-1\right)+\frac{7}{2{\pi }^2}\right)$

= $0+\frac{7}{2{\left(\frac{3}{2}\pi \right)}^2}-\left(4+\frac{7}{2{\pi }^2}\right)$

= $\frac{14}{9{\pi }^2}-\left(4+\frac{7}{2{\pi }^2}\right)$

= $-4-\frac{7}{2{\pi }^2}+\frac{14}{9{\pi }^2}$

= $-4-\frac{35}{18{\pi }^2}$

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

$=$ $3e^{-4+3}-3e^{-1+3}$

= $3e^{-1}-3e^2$