6.1 Areas Between Curves/22

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= \sin(\frac{\pi x}{2})} & \color{royalblue}\mathbf{y=x} \\ \end{align} }

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\sin({\frac {x\pi }{2}})&=x\\x&=0\\x&=1\\\end{aligned}}}

$\int _{0}^{1}\left(\sin \left({\frac {x\pi }{2}}\right)-x\right)dx$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{0}^{1}\left(\sin \left({\frac {x\pi }{2}}\right)\right)dx\\u={\frac {x\pi }{2}}\\du={\frac {\pi }{2}}dx\\{\frac {2}{\pi }}du=dx\\\end{aligned}}}

New upper limit: ${\frac {(0)\pi }{2}}=0$

New lower limit: Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {(1)\pi }{2}}={\frac {\pi }{2}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{0}^{1}\left(\sin \left({\frac {x\pi }{2}}\right)\right)dx&={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\sin(u)du\\&={\frac {2}{\pi }}\left[-\cos(u)\right]{\Bigg |}_{0}^{\frac {\pi }{2}}\\&={\frac {2}{\pi }}\left[-\cos({\frac {\pi }{2}})+\cos(0)\right]&={\frac {2}{\pi }}[0+1]\end{aligned}}}