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?