6.1 Areas Between Curves/12

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

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

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

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 \int_{0}^{2} \left[(4x-x^2) - (x^2)\right]dx = \int_{0}^{2} (4x)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} &= \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{align} }