6.1 Areas Between Curves/22

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

${\begin{aligned}\sin({\frac {x\pi }{2}})&=x\\x&=0\\x&=1\\\end{aligned}}$

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

${\begin{aligned}\int _{0}^{1}\left(\sin({\frac {x\pi }{2}})\right)dx&=\int _{0}^{\frac {\pi }{2}}\sin(u)du\\&u={\frac {x\pi }{2}}\\&du={\frac {\pi }{2}}dx\\&{\frac {2}{\pi }}du=dx\\\end{aligned}}$

New upper limit: ${\frac {(0)\pi }{2}}=0$ New lower limit: ${\frac {(1)\pi }{2}}={\frac {\pi }{2}}$