$$\underset{3}{\overset{7}{\int }}\mathrm{f(x)}dx$$ 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 = $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ : Dreiecksfläche I₂ = $\frac{\mathrm{(5\; -\; 3)\; \cdot \; <mo>(</mo>\; <mo>-</mo>\; <mn>4</mn>\; <mo>)</mo>}}{2}$ = $\frac{\mathrm{-8}}{2}$ = -4.

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

Somit gilt:

$$\underset{3}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ = $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ + $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = -4 +3 = -1

= ${\left[-x^3+x^2\right]}_{-3}^1$

$=$ $-1^3+1^2-\left(-{\left(-3\right)}^3+{\left(-3\right)}^2\right)$

= $-1+1-\left(-\left(-27\right)+9\right)$

= $-1+1-\left(27+9\right)$

= $0-1·36$

= $0-36$

= $-36$

= ${\left[\frac{2}{3}{x}^{\frac{3}{2}}-\frac{1}{2}{x}^{-2}\right]}_4^9$

= ${\left[\frac{2}{3}{\left(\sqrt{x}\right)}^3-\frac{1}{2x^2}\right]}_4^9$

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

= $\frac{2}{3}\cdot 3^3-\frac{1}{2}\cdot \left(\frac{1}{81}\right)-\left(\frac{2}{3}\cdot 2^3-\frac{1}{2}\cdot \left(\frac{1}{16}\right)\right)$

= $\frac{2}{3}\cdot 27-\frac{1}{162}-\left(\frac{2}{3}\cdot 8-\frac{1}{32}\right)$

= $18-\frac{1}{162}-\left(\frac{16}{3}-\frac{1}{32}\right)$

= $\frac{2916}{162}-\frac{1}{162}-\left(\frac{512}{96}-\frac{3}{96}\right)$

= $\frac{2915}{162}-1·\frac{509}{96}$

= $\frac{2915}{162}-\frac{509}{96}$

= $\frac{46640}{2592}-\frac{13743}{2592}$

= $\frac{2915}{162}-\frac{509}{96}$

= $\frac{46640}{2592}-\frac{13743}{2592}$

= $\frac{32897}{2592}$

= ${\left[3\cdot \sin \left(x-\frac{3}{2}\pi \right)\right]}_0^{\pi }$

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

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

= $3\cdot \left(-1\right)-3\cdot 1$

= $-3-3$

= $-6$

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

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

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

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

= $\frac{4}{3}{e}^{\frac{9}{2}\pi }-4-\frac{4}{3}{e}^{\frac{3}{2}\pi }-1·4$

= $\frac{4}{3}{e}^{\frac{9}{2}\pi }-4-\frac{4}{3}{e}^{\frac{3}{2}\pi }-4$

= $\frac{4}{3}{e}^{\frac{9}{2}\pi }-\frac{4}{3}{e}^{\frac{3}{2}\pi }-4-4$

= $\frac{4}{3}{e}^{\frac{9}{2}\pi }-\frac{4}{3}{e}^{\frac{3}{2}\pi }-8$

= ${\left[-\frac{1}{6}{\left(3x-5\right)}^4+5x\right]}_0^2$

$=$ $-\frac{1}{6}\cdot {\left(3\cdot 2-5\right)}^4+5\cdot 2-\left(-\frac{1}{6}\cdot {\left(3\cdot 0-5\right)}^4+5\cdot 0\right)$

= $-\frac{1}{6}\cdot {\left(6-5\right)}^4+10-\left(-\frac{1}{6}\cdot {\left(0-5\right)}^4+0\right)$

= $-\frac{1}{6}\cdot 1^4+10-\left(-\frac{1}{6}\cdot {\left(-5\right)}^4+0\right)$

= $-\frac{1}{6}\cdot 1+10-\left(-\frac{1}{6}\cdot 625+0\right)$

= $-\frac{1}{6}+10-\left(-\frac{625}{6}+0\right)$

= $-\frac{1}{6}+\frac{60}{6}-\left(-\frac{625}{6}+0\right)$

= $\frac{59}{6}+\frac{625}{6}$

= $114$