y = x 2 y = 4 x − x 2 x = 0 x = 2 {\displaystyle {\begin{aligned}&\color {red}\mathbf {y=x^{2}} &\color {royalblue}\mathbf {y=4x-x^{2}} \\&x=0&x=2\\\end{aligned}}}
∫ 0 2 [ ( 4 x − x 2 ) − ( x 2 ) ] d x {\displaystyle \int _{0}^{2}\left[(4x-x^{2})-(x^{2})\right]dx}
4 − x 2 = x 2 4 x − 2 x 2 = 0 2 x ( 2 − x ) = 0 x = 0 x = 2 {\displaystyle {\begin{aligned}4-x^{2}&=x^{2}\\4x-2x^{2}&=0\\2x(2-x)&=0\\x&=0&x=2\end{aligned}}}
∫ 0 2 [ ( 4 x − x 2 ) − ( x 2 ) ] d x = ∫ 0 2 ( 4 x ) d x {\displaystyle \int _{0}^{2}\left[(4x-x^{2})-(x^{2})\right]dx=\int _{0}^{2}(4x)dx}
= [ 4 x 2 2 ] | 0 2 = [ 2 x 2 ] | 0 2 = [ 2 ( 2 ) 2 ] − [ 2 ( 0 ) 2 ] = 8 − 0 = 8 {\displaystyle {\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}}}
![{\displaystyle \int _{0}^{2}\left[(4x-x^{2})-(x^{2})\right]dx}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6d1ebe4106c00946ea1b68c9603574e2f1fdbcec)
![{\displaystyle \int _{0}^{2}\left[(4x-x^{2})-(x^{2})\right]dx=\int _{0}^{2}(4x)dx}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0406095dc5c7b6b0a0ae7a38f759508615fe8bb2)
![{\displaystyle {\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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f4fe3a4714b1b9304b6b82d1c41558f4831928be)
