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

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

Somit gilt:

$$\underset{3}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = I₂ + I₃ = $$\underset{3}{\overset{5}{\int }}\mathrm{f(x)}dx$$ + $$\underset{5}{\overset{7}{\int }}\mathrm{f(x)}dx$$ = -2 -4 = -6

= ${\left[\frac{3}{2}{x}^2+5x\right]}_0^3$

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

= $\frac{3}{2}\cdot 9+15-\left(\frac{3}{2}\cdot 0+0\right)$

= $\frac{27}{2}+15-\left(0+0\right)$

= $\frac{27}{2}+\frac{30}{2}+0$

= $\frac{57}{2}$

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

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

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

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

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

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

= $-9-\frac{9}{2}\cdot {\pi }^3-9+\frac{1}{6}\cdot {\pi }^3$

= $-9-9-\frac{9}{2}\cdot {\pi }^3+\frac{1}{6}\cdot {\pi }^3$

= $-18-\frac{13}{3}\cdot {\pi }^3$

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

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

= $e^{-12+3}-e^{-3+3}$

= $e^{-9}-e^0$

= $e^{-9}-1$

= ${\left[-\frac{7}{4}\cdot \sin \left(x\right)-\frac{7}{6}{e}^{3x}\right]}_{\frac{1}{2}\pi }^{\pi }$

$=$ $-\frac{7}{4}\cdot \sin \left(\pi \right)-\frac{7}{6}{e}^{3\cdot \pi }-\left(-\frac{7}{4}\cdot \sin \left(\frac{1}{2}\pi \right)-\frac{7}{6}{e}^{3\cdot \left(\frac{1}{2}\pi \right)}\right)$

= $-\frac{7}{4}\cdot 0-\frac{7}{6}{e}^{3\cdot \pi }-\left(-\frac{7}{4}\cdot 1-\frac{7}{6}{e}^{3\cdot \left(\frac{1}{2}\pi \right)}\right)$

= $0-\frac{7}{6}{e}^{3\cdot \pi }-\left(-\frac{7}{4}-\frac{7}{6}{e}^{3\cdot \left(\frac{1}{2}\pi \right)}\right)$

= $-\frac{7}{6}{e}^{3\pi }-\left(-\frac{7}{6}{e}^{\frac{3}{2}\pi }-\frac{7}{4}\right)$

= $-\frac{7}{6}{e}^{3\pi }+\frac{7}{6}{e}^{\frac{3}{2}\pi }-1·\left(-\frac{7}{4}\right)$

= $-\frac{7}{6}{e}^{3\pi }+\frac{7}{6}{e}^{\frac{3}{2}\pi }+\frac{7}{4}$

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

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

= $\frac{2}{3}\cdot \cos \left(-6\pi \right)-\frac{2}{3}\cdot \cos \left(-\frac{3}{2}\pi \right)$

= $\frac{2}{3}\cdot 1-\frac{2}{3}\cdot 0$

= $\frac{2}{3}+0$

= $\frac{2}{3}+0$

= $\frac{2}{3}$