6.1 Areas Between Curves/23

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} }

$\int _{0}^{\frac {\pi }{6}}\left(\cos(x)-\sin(2x)\right)dx+\int _{\frac {\pi }{6}}^{\frac {\pi }{2}}\left(\sin(2x)-\cos(x)\right)dx$

${\begin{aligned}\int _{0}^{\frac {\pi }{6}}\left(\cos(x)-\sin(2x)\right)dx&=\left[\sin(x)+{\frac {1}{2}}\cos(2x)\right]{\Bigg |}_{0}^{\frac {\pi }{6}}\\[2ex]&=\left[\sin({\frac {\pi }{6}})+{\frac {1}{2}}\cos({\frac {2\pi }{6}})\right]-\left[\sin(0)+{\frac {1}{2}}\cos(2(0))\right]\\[2ex]&=\left[{\frac {1}{2}}+{\frac {1}{2}}\left({\frac {1}{2}}\right)\right]-\left[0-{\frac {1}{2}}(1)\right]\\[2ex]&={\frac {1}{2}}+{\frac {1}{4}}-\left[0+{\frac {1}{2}}\right]\\&={\frac {1}{4}}\end{aligned}}$

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_{\frac{\pi}{6}}^{\frac{\pi}{2}} \left[\sin(2x)-\cos(x)\right]dx \end{align} }