6.1 Areas Between Curves/17

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} & \color{red}\mathbf{y=\sqrt{x},\ y=\frac{1}{2}x,\ x=9 \\[2ex] & \sqrt{x}=\frac{1}{2}x\ \rightarrow \ \sqrt{x}-\frac{1}{2}x=0\ \rightarrow \ \sqrt{4}\ -\frac{1}{2}(4) = 2-2=0,\ x=4 \\[2ex] & A= \int_{0}^{4} \left[\sqrt{x} - \frac{1}{2}x \right]\mathrm{d}x + \int_{4}^{9} \left[ \frac{1}{2}x - \sqrt{x} \right]\mathrm{d}x \\[2ex] &= \ \left[\frac{2}{3}x^\frac{3}{2} - \frac{1}{4}x^2 \right]_{0}^{4} \ + \ \left[ \frac{1}{4}x^2 - \frac{2}{3}x^\frac{3}{2} \right]_{4}^{9} \\[2ex] &=\left[ \frac{2}{3} \left(4\right)^\frac{3}{2} - \frac{1}{4} \left(4\right)^2 \right] - \left[0\right] + \left[\frac{1}{4}\left(9\right)^2 - \frac{2}{3}\left(9\right)^\frac{3}{2} \right] - \left[\frac{1}{4}\left(4\right)^2 - \frac{2}{3}\left(4\right)^\frac{3}{2} \right] = \left[ \frac{16}{3} - 4 \right] - \left[0\right] + \left[\frac{81}{4} - 18\right] - \left[4 - \frac{16}{3}\right] \\[2ex] &=\frac{4}{3} + \frac{9}{4} + \frac{4}{3} = \frac{8}{3} + \frac{9}{4} = \frac{32}{12} + \frac{27}{12} \\[2ex] &= \frac{59}{12} \end{align} }