Area Between Curves: Find the Region Between y=sqrt(x) & y=1/2x

[3.141592654]

Homework Statement

Find the area of the region between the two curves.

$$y=\sqrt{x}$$

$$y=\frac{1}{2}x$$

$$x=9$$

Homework Equations

The Attempt at a Solution

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

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

$$=\frac{1}{2}\intxdx-\int x^\frac{1}{2}dx$$

$$=\frac{1}{2}\frac{x^2}{2}-\frac{2}{3}x^\frac{3}{2}$$

$$=\frac{1}{4}(9^2-4^2)-\frac{2}{3}(9^\frac{3}{2}-4^\frac{3}{2})$$

$$=\frac{1}{4}(65)-\frac{2}{3}(27-8)$$

$$=\frac{1}{4}(65)-\frac{2}{3}(19)$$

$$=\frac{65}{4}-\frac{38}{3}$$

$$=\frac{195-152}{12}$$

$$=\frac{43}{12}$$

The answer in the book is $$\frac{59}{12}$$.

 

[Dick]

It looks to me like there are two parts to the region lying between the two curves. What about the [0,4] part? Shouldn't you add the areas of both of them?