Answer
$ \frac{10}{3}-\frac{4\sqrt[]{2}}{3} $
Work Step by Step
Step 1 of 3:
First, find the points of intersection of the curves $ y=\sqrt[]{x} $
and $ y=x^{2} $
before
$ x=2 $
$ x^{2}=\sqrt[]{x} $
$ x^{4}-x=0 $
or $ x=0 $
or $ x=1 $
The points of intersection are (0, 0) and (1, 1).
Step 2 of 3:
Sketch the curves.
Find the area of the shaded region.
Step 3 of 3:
Area of the shaded region $ A= \int _{0}^{1} \left( \sqrt[]{x}-x^{2} \right) dx+ \int _{1}^{2} \left( x^{2}-\sqrt[]{x} \right) dx $
$ A= \int _{0}^{1} \left( x^{\frac{1}{2}}-x^{2} \right) dx+ \int _{1}^{2} \left( x^{2}-x^{\frac{1}{2}} \right) dx $
$ A= \left[ \frac{2}{3}x^{\frac{3}{2}}-\frac{1}{3}x^{3} \right] _{0}^{1}+ \left[ \frac{1}{3}x^{3}-\frac{2}{3}x^{\frac{3}{2}} \right] _{1}^{2} $
By the fundamental
Theorem of calculus
$ A= \left[ \frac{2}{3}-\frac{1}{3} \right] - \left[ \frac{8}{3}-\frac{4}{3}\sqrt[]{2}-\frac{1}{3}+\frac{2}{3} \right] $
$ A=\frac{10}{3}-\frac{4\sqrt[]{2}}{3} $
