6.1 Areas Between Curves/23

${\begin{aligned}&\color {red}\mathbf {y=\cos(x)} &\color {royalblue}\mathbf {y=\sin(2x)} \\&x=0&x={\frac {\pi }{2}}\\\end{aligned}}$

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} \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 = \frac{1}{4}+\frac{1}{4} = \frac{2}{4} = \frac{1}{2} }

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}-\frac{1}{2} \\ &= \frac{1}{4} \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} \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}} \\ [2ex] &= \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] \\[2ex] &= \left[-\frac{1}{2}\left(-1\right)-1\right]-\left[-\frac{1}{2}\left(\frac{1}{2}\right)-\frac{1}{2}\right]\\ &= \frac{1}{2}-1+\frac{1}{4}+\frac{1}{2} \\ &= \frac{1}{4} \end{align} }