Answer
The area of the shaded region is $5/6$.
Work Step by Step
With respect to $y$, the formulas of the curves are
- Curve $y=x$: $x=y$
- Curve $y=\frac{x^2}{4}$: $x=\pm\sqrt{4y}=\pm2\sqrt y=\pm2y^{1/2}$
However, from the graph, we can see that we work only with the positive values of $x$, so $x=2y^{1/2}$.
In this exercise, it might be easier if we carry out integration with respect to $y$ by the formula $$A=\int^d_c[f(y)-g(y)]dy$$ to calculate the area, in which $f$ denotes the right-hand curve and $g$ the left-hand curve.
Here, the shaded region is bounded on the right by the curve $x=2y^{1/2}$ and on the left by the curve $x=y$ and runs from $y=0$ to $y=1$.
Therefore, the area of the shaded region is $$A=\int^{1}_{0}(2y^{1/2}-y)dy$$ $$A=\Big(\frac{2y^{3/2}}{\frac{3}{2}}-\frac{y^2}{2}\Big)\Big]^{1}_{0}=\Big(\frac{4y^{3/2}}{3}-\frac{y^2}{2}\Big)\Big]^{1}_{0}$$ $$A=\frac{4}{3}-\frac{1}{2}=\frac{5}{6}$$
