- #1

3.141592654

- 85

- 0

## Homework Statement

Find the area of the region between the two curves.

[tex]y=\sqrt{x}[/tex]

[tex]y=\frac{1}{2}x[/tex]

[tex]x=9[/tex]

## Homework Equations

## The Attempt at a Solution

The domain of the region is [4,9]:

[tex]\int\frac{1}{2}x-\sqrt{x}dx[/tex] with limits of integration [tex][4, 9][/tex]

[tex]=\frac{1}{2}\intxdx-\int x^\frac{1}{2}dx[/tex]

[tex]=\frac{1}{2}\frac{x^2}{2}-\frac{2}{3}x^\frac{3}{2}[/tex]

[tex]=\frac{1}{4}(9^2-4^2)-\frac{2}{3}(9^\frac{3}{2}-4^\frac{3}{2})[/tex]

[tex]=\frac{1}{4}(65)-\frac{2}{3}(27-8)[/tex]

[tex]=\frac{1}{4}(65)-\frac{2}{3}(19)[/tex]

[tex]=\frac{65}{4}-\frac{38}{3}[/tex]

[tex]=\frac{195-152}{12}[/tex]

[tex]=\frac{43}{12}[/tex]

The answer in the book is [tex]\frac{59}{12}[/tex].