Question:

<div class='solution'> <p>Intersections: \(y=4x-2\) and \(y^2=4x\): \((4x-2)^2=4x\Rightarrow16x^2-20x+4=0\Rightarrow4x^2-5x+1=0\Rightarrow(4x-1)(x-1)=0\Rightarrow x=1/4,1\).</p> <p>At \(x=1/4\): \(y=-1\). At \(x=1\): \(y=2\). Integrate w.r.t. y from −1 to 2:</p> <p>\[A=\int_{-1}^2\left[\frac{y^2}{4}-\frac{y+2}{4}\right]dy=\frac{1}{4}\int_{-1}^2(y^2-y-2)dy=\frac{1}{4}\left[\frac{y^3}{3}-\frac{y^2}{2}-2y\right]_{-1}^2\]</p> <p>\(=\frac{1}{4}\left[\left(\frac{8}{3}-2-4\right)-\left(-\frac{1}{3}-\frac{1}{2}+2\right)\right]=\frac{1}{4}\cdot\frac{(-9)}{...}=\frac{9}{8}\). ✓</p> </div>