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<tex> | |||
T = \int_{A}r(\tau{dA})= | |||
\int_{A}r\left(\frac{r}{R}\tau_{max}\right)dA | |||
& = & \frac{\tau_{max}}{R}\int_{A}r^{2}dA = | |||
\frac{\tau_{max}}{R}\int_{0}^{2\pi}\!\int_{0}^{R}r^{2}r{dr}{d\phi} | |||
= | |||
\frac{\tau_{max}}{R}\int_{0}^{2\pi}\!\int_{0}^{R}r^{3}{dr}{d\phi} | |||
=& = | |||
\frac{\tau_{max}}{R}\int_{0}^{2\pi}\left[\frac{r^4}{4}\right]_{r=0}^{r=R}{d\phi} | |||
= \frac{\tau_{max}}{R}\int_{0}^{2\pi}\frac{R^4}{4}{d\phi} | |||
& = & | |||
\frac{\tau_{max}}{R}\left[\frac{R^4}{4}\phi\right]_{\phi=0}^{\phi=2\pi} | |||
= \frac{\tau_{max}}{R}\times\frac{\pi{R}^4}{2} = | |||
\frac{\tau_{max}}{R}\times{J} | |||
</tex> | |||
[[Category:Software/Calculi]] | [[Category:Software/Calculi]] | ||
Revision as of 02:03, 16 February 2009
x > y
<tex> T = \int_{A}r(\tau{dA})= \int_{A}r\left(\frac{r}{R}\tau_{max}\right)dA & = & \frac{\tau_{max}}{R}\int_{A}r^{2}dA = \frac{\tau_{max}}{R}\int_{0}^{2\pi}\!\int_{0}^{R}r^{2}r{dr}{d\phi} = \frac{\tau_{max}}{R}\int_{0}^{2\pi}\!\int_{0}^{R}r^{3}{dr}{d\phi}
&
\frac{\tau_{max}}{R}\int_{0}^{2\pi}\left[\frac{r^4}{4}\right]_{r=0}^{r=R}{d\phi} = \frac{\tau_{max}}{R}\int_{0}^{2\pi}\frac{R^4}{4}{d\phi} & = & \frac{\tau_{max}}{R}\left[\frac{R^4}{4}\phi\right]_{\phi=0}^{\phi=2\pi}
\frac{\tau_{max}}{R}\times\frac{\pi{R}^4}{2}
\frac{\tau_{max}}{R}\times{J} </tex>