6.1 Areas Between Curves/23

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

${\begin{aligned}\cos(x)&=\sin(2x)\\x&={\frac {\pi }{2}}\\x&={\frac {\pi }{6}}\\\end{aligned}}$

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

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] \end{align} }