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Math Help - Find the area of the region

  1. #1
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    Find the area of the region

    Find the area of the region
    <br />
\left \{ <br />
(x,y) : x^2 \le y \le |x|<br />
\right \}<br />
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  2. #2
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    Hello,

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

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

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

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  3. #3
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    why have u taken 2.....and how do u know it is symmetric ?????
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  4. #4
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    Quote Originally Posted by zorro View Post
    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.
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  5. #5
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    Quote Originally Posted by zorro View Post
    why have u taken 2.....and how do u know it is symmetric ?????
    The boundaries are given by y= x^2 and 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.
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