![\bf \begin{cases} y=sin\left( \frac{\pi x}{6} \right)\\ 0\le x\le 6\\ \textit{about the x-axis, or }y=0 \end{cases}\\\\ -----------------------------\\\\ \textit{using the disc method} \\\\ V=\int\limits_{0}^{6}\pi \left[ sin\left( \frac{\pi x}{6} \right) \right]^2\cdot dx\implies \pi\int\limits_{0}^{6} \left[ sin\left( \frac{\pi x}{6} \right) \right]^2\cdot dx \\\\\\ \pi\int\limits_{0}^{6} sin^2\left( \frac{\pi x}{6} \right) \cdot dx\\\\ -----------------------------\\\\](/tpl/images/0425/8134/bfb7b.png)
![\bf \pi\int\limits_{0}^{6} \left[ sin\left( \frac{\pi x}{6} \right) \right]^2\cdot dx\implies \pi\int\limits_{0}^{6} \cfrac{1-cos\left(2\cdot \frac{\pi x}{6} \right)}{2}\cdot dx \\\\\\ \pi\int\limits_{0}^{6}\cfrac{1}{2}dx-\pi \cdot \cfrac{1}{2}\pi\int\limits_{0}^{6}cos\left(\frac{\pi x}{3} \right)dx \\\\\\ \left[\cfrac{\pi x}{2}-\cfrac{\pi }{2}\cdot \cfrac{sin\left(\frac{\pi x}{3} \right)}{\frac{\pi x}{3} } \right]\implies \left[ \cfrac{\pi }{2}x-\cfrac{3sin\left(\frac{\pi x}{3} \right)}{2x} \right]_0^6](/tpl/images/0425/8134/6bc7f.png)
Find the exact area of the surface obtained by rotating the curve about the x-axis.
y = sin ((...
Mathematics, 19.12.2019 07:31 keke6361
Find the exact area of the surface obtained by rotating the curve about the x-axis.
y = sin ((pi*x)/6)
0 ≤ x ≤ 6
Answers: 1
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