# Thread: Find the area of the region

1. ## Find the area of the region

Find the area of the region
$\displaystyle \left \{ (x,y) : x^2 \le y \le |x| \right \}$

2. Hello,

You have to find the aera between the following functions $\displaystyle x\mapsto x^2$ and $\displaystyle x\mapsto x$ on $\displaystyle [0,1]$ (because curves meet for $\displaystyle x=1$).

By a symmetry the aera of the region is $\displaystyle \mathcal{A}=2\left(\int_{0}^{1}(x-x^2)dx\right)$

And you obtain : $\displaystyle \color{blue}\boxed{\mathcal{A}=\frac{1}{3}}$

3. why have u taken 2.....and how do u know it is symmetric ?????

4. Originally Posted by zorro
why have u taken 2.....and how do u know it is symmetric ?????
Draw the graphs of y = x^2 and y = |x|. The set up of the integrals is obvious.

5. Originally Posted by zorro
why have u taken 2.....and how do u know it is symmetric ?????
The boundaries are given by $\displaystyle y= x^2$ and $\displaystyle y= |x|$, both of which are "even functions" and symmetric about the y-axis.

And he multiplied by 2 because it is symmetric. The area from -1 to 0 is the same as the area from 0 to 1- to find the area from -1 to 1, find the area from 0 to 1 and multiply by 2.