Wiki-Quellcode von Lösung Volumenberechnung 2
Zuletzt geändert von Anna Kukin am 2026/08/03 19:45
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| author | version | line-number | content |
|---|---|---|---|
| 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 | Ebenso sind zur Berechnung die ersten beiden binomischen Formeln notwendig. | ||
| 3 | |||
| 4 | a) | ||
| 5 | {{formula}} | ||
| 6 | \begin{align*} | ||
| 7 | V &= \pi \cdot \int_{0}^{4} (x+9)^2 \, dx \\ | ||
| 8 | &= \pi \cdot \int_{0}^{4} (x^2 + 18x + 81) \, dx \\ | ||
| 9 | &= \pi \cdot \left[ \frac{1}{3}x^3 + 9x^2 + 81x \right]_{0}^{4} \\ | ||
| 10 | &= \pi \cdot \left( \left(\frac{1}{3} \cdot 4^3 + 9 \cdot 4^2 + 81 \cdot 4\right) - 0 \right) \\ | ||
| 11 | &= \pi \cdot \left( \frac{64}{3} + 144 + 324 \right) \\ | ||
| 12 | &= \pi \cdot \left( \frac{64}{3} + \frac{1404}{3} \right) \\ | ||
| 13 | &= \frac{1468}{3}\pi \approx 1537{,}29 \text{ VE} | ||
| 14 | \end{align*} | ||
| 15 | {{/formula}} | ||
| 16 | |||
| 17 | b) | ||
| 18 | {{formula}} | ||
| 19 | \begin{align*} | ||
| 20 | V &= \pi \cdot \int_{0}^{4} \left(x^2+7x\right)^2 \, dx \\ | ||
| 21 | &= \pi \cdot \int_{0}^{4} (x^4 + 14x^3 + 49x^2) \, dx \\ | ||
| 22 | &= \pi \cdot \left[ \frac{1}{5}x^5 + \frac{14}{4}x^4 + \frac{49}{3}x^3 \right]_{0}^{4} \\ | ||
| 23 | &= \pi \cdot \left( \left(\frac{1}{5} \cdot 4^5 + \frac{7}{2} \cdot 4^4 + \frac{49}{3} \cdot 4^3\right) - 0 \right) \\ | ||
| 24 | &= \pi \cdot \left( \frac{1024}{5} + \frac{7}{2} \cdot 256 + \frac{49}{3} \cdot 64 \right) \\ | ||
| 25 | &= \pi \cdot \left( \frac{1024}{5} + 896 + \frac{3136}{3} \right) \\ | ||
| 26 | &= \pi \cdot \left( \frac{6144}{30} + \frac{26880}{30} + \frac{31360}{30} \right) \\ | ||
| 27 | &= \frac{64384}{30}\pi \approx 6742{,}28 \text{ VE} | ||
| 28 | \end{align*} | ||
| 29 | {{/formula}} | ||
| 30 | |||
| 31 | c) | ||
| 32 | {{formula}} | ||
| 33 | \begin{align*} | ||
| 34 | V &= \pi \cdot \int_{0}^{4} \left(e^{2x}-1\right)^2 \, dx \\ | ||
| 35 | &= \pi \cdot \int_{0}^{4} (e^{4x} - 2e^{2x} + 1) \, dx \\ | ||
| 36 | &= \pi \cdot \left[ \frac{1}{4}e^{4x} - e^{2x} + x \right]_{0}^{4} \\ | ||
| 37 | &= \pi \cdot \left( \left(\frac{1}{4}e^{16} - e^{8} + 4\right) - \left(\frac{1}{4}e^{0} - e^{0} + 0\right) \right) \\ | ||
| 38 | &= \pi \cdot \left( \frac{1}{4}e^{16} - e^{8} + 4 - \left(\frac{1}{4} - 1\right) \right) \\ | ||
| 39 | &= \pi \cdot \left( \frac{1}{4}e^{16} - e^{8} + 4 - \left(-\frac{3}{4}\right) \right) \\ | ||
| 40 | &= \pi \cdot \left( \frac{1}{4}e^{16} - e^{8} + \frac{19}{4} \right) \approx 6\,969\,784{,}85 \text{ VE} | ||
| 41 | \end{align*} | ||
| 42 | {{/formula}} |