6.1 Areas Between Curves/12

${\begin{aligned}&\color {red}\mathbf {y=x^{2}} &\color {royalblue}\mathbf {y=4x-x^{2}} \\&x=0&x=2\\\end{aligned}}$

$\int _{0}^{2}\left[(4x-x^{2})-(x^{2})\right]dx$

${\begin{aligned}4-x^{2}&=x^{2}\\4x-2x^{2}&=0\\2x(2-x)&=0\\x&=0&x=2\end{aligned}}$

$\int _{0}^{2}\left[(4x-x^{2})-(x^{2})\right]dx=\int _{0}^{2}(4x)dx$

${\begin{aligned}&=\left[{\frac {4x^{2}}{2}}\right]{\Bigg |}_{0}^{2}\\[2ex]&=\left[2x^{2}\right]{\Bigg |}_{0}^{2}\\[2ex]&=\left[2(2)^{2}\right]-\left[2(0)^{2}\right]=8-0\\[2ex]&=8\end{aligned}}$