6.1 Areas Between Curves/12

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=x^2} & \color{royalblue}\mathbf{y=4x-x^2} \\ & x=0 & x=2 \\ \end{align} }

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}

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

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[16-\frac{16}{3}\right]-\left[-16+\frac{16}{3}\right] = 32-\frac{32}{3} \\[2ex] &= \frac{64}{3} \end{align} }