6.1 Areas Between Curves/19
19)
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{x=4+y^2} & \color{royalblue}\mathbf{x=2y^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 \begin{align} 4+y^2 &= 2y^2 \\ 4 &=y^2 \\ y &=\sqrt{4} \\ y &= \pm2 \\[2ex] \int_{-2}^{2} [(4+y^2)-(2y^2)]dy \\ =\int_{-2}^{2} [4-y^2]dy \\[2ex] =\left[4y-\frac{y^3}{3}\right]\Bigg|_{-2}^{2}\\[2ex] =4(2)-\frac{(2)^3}{3}-(4(-2)-\frac{(-2)^3}{3})\\[2ex] =8-\frac{8}{3}-(-8+\frac{8}{3})\\[2ex] =16-\frac{16}{3}\\[2ex] =\frac{48}{3}-\frac{16}{3}\\[2ex] =\frac{32}{3}\\ \end{align} }