# Thread: Area of region bounded by sqrt x, 2 - sqrt x and y = 0

1. ## Area of region bounded by sqrt x, 2 - sqrt x and y = 0

I need to find the area of the region bounded by the graphs of:
y = sqrt x,
y = 2 - sqrt x, and
y = 0.

I have sketched the graph and I know where the boundary is.

I intend to integrate with respect to y. Am I on the right track?
$\displaystyle \int_0^2 y^2 - (4 - y^2) dy$

The answer given is 4 sq units but I don't think I will get that answer after integration.

2. Hello, darkvelvet!

And I don't agree with their answer.

$\displaystyle \text{Find the area of the region bounded by: }\;\begin{Bmatrix}y \:=\:\sqrt{x} \\ y \:=\:2 - \sqrt{x} \\ y\:=\:0.\end{Bmatrix}$

The graph of $\displaystyle y = \sqrt{x}$ is the upper half of a "horizontal" parabola,
. . vertex at (0,0), opening to the right.

The graph of $\displaystyle y =-\sqrt{x}$ is the above graph reflected over the $\displaystyle x$-axis.
The graph of $\displaystyle y \,=\,2 - \sqrt{x}$ is the same graph raised 2 units.

And $\displaystyle \,y = 0$ is the $\displaystyle x$-axis.

The graph looks like this:

Code:
      |
2 *                               *
|                     *
|*            *
| *     *
|   *.
| .:|:::*..
|*::|:::::::::*...
|:::|:::::::::::::::::*.....
- - * - + - - - - - - - - - - - - - * - -
|   1                           4
|

$\displaystyle \displaystyle \text{Area} \;=\;\int^1_0\!\! x^{\frac{1}{2}}\,dx + \int^4_1\!\!\left(2-x^{\frac{1}{2}}\right)\,dx$

$\displaystyle \displaystyle A = \int_0^1 (2-y)^2 - y^2 \, dy = 2$