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} }

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

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] &= \frac{1}{2}+\frac{1}{2}(\frac{1}{2}) &= \frac{1}{2}+\frac{1}{4}-(0+\frac{1}{2}) \end{align} }