Chapter 6.4, Problem 46E

To determine

The area of the shaded region.

Answer to Problem 46E

For shaded region,

  $\text{Total}\text{Area}=3$

Explanation of Solution

Given information:

  

Split the graph into two portions at the point [1, 1].

We are required to deal with each portion separately.

First portion $\left(y=\sqrt{x}\right)$ :

  ${\displaystyle {\int }_0^1\sqrt{x}dx}={\displaystyle {\int }_0^1{x}^{1/2}dx}={\frac{2}{3}{x}^{3/2}|}_0^1=\frac{2}{3}{\left(1\right)}^{3/2}-\frac{2}{3}{\left(0\right)}^{3/2}=\frac{2}{3}$

Second portion $\left(y=x^2\right)$ :

  ${\displaystyle {\int }_1^2{x}^{2}dx}={\frac{1}{3}{x}^3|}_1^2=\frac{1}{3}{\left(2\right)}^3-\frac{1}{3}{\left(1\right)}^3=\frac{8}{3}-\frac{1}{3}=\frac{7}{3}$

Now,

We have the areas of both first and second portion.

Add up the areas of both first portion and second portion to get the total area of the shaded region:

  $\text{Total}\text{Area}={\displaystyle {\int }_0^1\sqrt{x}dx}+{\displaystyle {\int }_1^2{x}^{2}dx}=\frac{2}{3}+\frac{7}{3}=\frac{9}{3}=3$