Die Funktion $k(x)k(x)$ ist eine Ursprungsgerade mit der Steigung $m={\displaystyle \frac{r}{h}}m=\backslash dfrac\{r\}\{h\}$.

Damit gilt: $k(x)={\displaystyle \frac{r}{h}}\cdot xk(x)=\backslash dfrac\{r\}\{h\}\backslash cdot\; x$

Das Rotationsvolumen berechnest du mit der Formel:

$V_K=\pi \cdot {\int }_0^h{(k(x))}^2\mathrm{d}x\Rightarrow V\_K=\backslash pi\backslash cdot\backslash int\_\{0\}^\{h\}\backslash left(k(x)\backslash right)^2\; \backslash mathrm\{d\}x\backslash ;\backslash Rightarrow\backslash ;$

$V_K=\pi \cdot {\int }_0^h{({\displaystyle \frac{r}{h}}\cdot x)}^2\mathrm{d}x\Rightarrow V\_K=\backslash pi\backslash cdot\backslash int\_\{0\}^\{h\}\backslash left(\backslash dfrac\{r\}\{h\}\backslash cdot\; x\backslash right)^2\; \backslash mathrm\{d\}x\backslash ;\backslash Rightarrow\backslash ;$

$V_K={\displaystyle \frac{\pi \cdot r^2}{h^2}}\cdot {\int }_0^h{x}^2\mathrm{d}xV\_K=\backslash dfrac\{\backslash pi\backslash cdot\; r^2\}\{h^2\}\backslash cdot\backslash int\_0^hx^2\backslash mathrm\{d\}x$

$V_K={\displaystyle \frac{\pi \cdot r^2}{h^2}}\cdot {\left[{\displaystyle \frac{1}{3}}{x}^3\right]}_0^{h}V\_K=\backslash dfrac\{\backslash pi\backslash cdot\; r^2\}\{h^2\}\backslash cdot\backslash left[\backslash dfrac\{1\}\{3\}x^3\backslash right]\_0^\{h\}$

$V_K={\displaystyle \frac{\pi \cdot r^2}{h^2}}\cdot ({\displaystyle \frac{1}{3}}{h}^3-0)={\displaystyle \frac{1}{3}}\cdot \pi r^{2}hV\_K=\backslash dfrac\{\backslash pi\backslash cdot\; r^2\}\{h^2\}\backslash cdot\backslash left(\backslash dfrac\{1\}\{3\}h^3-0\backslash right)=\backslash dfrac\{1\}\{3\}\backslash cdot\backslash pi\; r^2\; h$