Wiki-Quellcode von Lösung Volumenberechnung 1
Zuletzt geändert von Anna Kukin am 2026/08/03 19:35
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| author | version | line-number | content |
|---|---|---|---|
 |
1.1 | 1 | Zur Berechnung des Volumens verwenden wir die Formel {{formula}}V=\pi \cdot \int_{a}^{b} \left(f(x)\right)^2 \, dx{{/formula}} aus der Merkhilfe. |
| 2 | |||
| 3 | a) | ||
| 4 | {{formula}} | ||
| 5 | \begin{align*} | ||
| 6 | V &= \pi \cdot \int_{0}^{2} \left(\sqrt{8x+1}\right)^2 \, dx \\ | ||
| 7 | &= \pi \cdot \int_{0}^{2} (8x+1) \, dx \\ | ||
| 8 | &= \pi \cdot \left[ 4x^2 + x \right]_{0}^{2} \\ | ||
| 9 | &= \pi \cdot \left( (4 \cdot 2^2 + 2) - (4 \cdot 0^2 + 0) \right) \\ | ||
| 10 | &= \pi \cdot (16 + 2 - 0) \\ | ||
| 11 | &= 18\pi \approx 56{,}55 \text{ VE} | ||
| 12 | \end{align*} | ||
| 13 | {{/formula}} | ||
| 14 | |||
| 15 | b) | ||
| 16 | {{formula}} | ||
| 17 | \begin{align*} | ||
| 18 | V &= \pi \cdot \int_{0}^{2} \left(x^2\right)^2 \, dx \\ | ||
| 19 | &= \pi \cdot \int_{0}^{2} x^4 \, dx \\ | ||
| 20 | &= \pi \cdot \left[ \frac{1}{5}x^5 \right]_{0}^{2} \\ | ||
| 21 | &= \pi \cdot \left( \frac{1}{5} \cdot 2^5 - \frac{1}{5} \cdot 0^5 \right) \\ | ||
| 22 | &= \pi \cdot \left( \frac{32}{5} - 0 \right) \\ | ||
| 23 | &= \frac{32}{5}\pi = 6{,}4\pi \approx 20{,}11 \text{ VE} | ||
| 24 | \end{align*} | ||
| 25 | {{/formula}}))) |