6.1 Areas Between Curves/23: Difference between revisions
No edit summary |
No edit summary |
||
| Line 41: | Line 41: | ||
\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \left[\sin(2x)-\cos(x)\right]dx &= \left[-\frac{1}{2}\cos(2x) - \sin(x) \right]\Bigg|_{\frac{\pi}{6}}^{\frac{\pi}{2}} \\ | \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \left[\sin(2x)-\cos(x)\right]dx &= \left[-\frac{1}{2}\cos(2x) - \sin(x) \right]\Bigg|_{\frac{\pi}{6}}^{\frac{\pi}{2}} \\ | ||
&= \left[-\frac{1}{2}\cos(\frac{2\pi}{2})-sin(\frac{\pi}{2})\right] - \left[-\frac{1}{2}\cos | &= \left[-\frac{1}{2}\cos(\frac{2\pi}{2})-sin(\frac{\pi}{2})\right] - \left[-\frac{1}{2}\cos(\frac{2\pi}{6}) - \sin(\frac{\pi}{6})\right] | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Revision as of 02:03, 20 September 2022
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \color{red} \mathbf{y=\cos(x)} & \color{royalblue}\mathbf{y=\sin(2x)} \\ & x=0 & x=\frac{\pi}{2}\\ \end{align} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cos(x) &= \sin(2x) \\ x &= \frac{\pi}{2} \\ x &= \frac{\pi}{6} \\ \end{align} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\frac{\pi}{6}} \left(\cos(x) - \sin(2x) \right)dx + \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \left(\sin(2x)- \cos(x) \right)dx }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{0}^{\frac{\pi}{6}} \left(\cos(x) - \sin(2x) \right)dx &= \left[\sin(x)+\frac{1}{2}\cos(2x) \right]\Bigg|_{0}^{\frac{\pi}{6}} \\[2ex] &= \left[\sin(\frac{\pi}{6})+\frac{1}{2}\cos(\frac{2\pi}{6})\right]-\left[\sin(0)+\frac{1}{2}\cos(2(0))\right] \\[2ex] &= \left[\frac{1}{2}+\frac{1}{2}\left(\frac{1}{2}\right)\right]-\left[0-\frac{1}{2} (1)\right] \\[2ex] &= \frac{1}{2}+\frac{1}{4}-\left[0+\frac{1}{2}\right] \\ &= \frac{1}{4} \end{align} }
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{\frac {\pi }{6}}^{\frac {\pi }{2}}\left[\sin(2x)-\cos(x)\right]dx&=\left[-{\frac {1}{2}}\cos(2x)-\sin(x)\right]{\Bigg |}_{\frac {\pi }{6}}^{\frac {\pi }{2}}\\&=\left[-{\frac {1}{2}}\cos({\frac {2\pi }{2}})-sin({\frac {\pi }{2}})\right]-\left[-{\frac {1}{2}}\cos({\frac {2\pi }{6}})-\sin({\frac {\pi }{6}})\right]\end{aligned}}}