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Math Help - Area of region enclosed by lines and curves?

  1. #1
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    Area of region enclosed by lines and curves?

    Please help with this math problem; I’ve been working on it for weeks (literally). How do I solve it?

    Find the area of the regions enclosed by the lives and curves.


    14. y = x^4 – 4x^2 + 4 and y = x^2


    I've begun to work it out somewhat:
    ∫_(-1)^(-2) [x^2-(x^k-4x^2+4) ]ⅆx + ∫_1^2 [x_^4-4x^2+4-(x^2 ) ]ⅆx

    The rest I've made effort to work is too long to post; I got 0 as an answer which I'm sure isn't correct.
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  2. #2
    Behold, the power of SARDINES!
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    MHF.pdf

    I could not get my Latex to compile so here it is. It is also in the .pdf

    Code:
    \documentclass{amsart}
    \usepackage{amsmath}%
    \usepackage{amsfonts}%
    \usepackage{amssymb}%
    \usepackage{graphicx}
    \begin{document}
    \begin{equation*} \int_{-2}^{-1} {x}^{2}-({x}^{4}-4{x}^{2}+4)dx +
    \int_{-1}^{1} ({x}^{4}-4{x}^{2}+4)-{x}^{2}dx +
    \int_{1}^{2} {x}^{2}-({x}^{4}-4{x}^{2}+4)dx \end{equation*}
    
    Or by using symmetry this gives (Both functions are even)
    
    \begin{equation*} 2 \left[ \int_{0}^{1} ({x}^{4}-4{x}^{2}+4)-{x}^{2}dx +
    \int_{1}^{2} {x}^{2}-({x}^{4}-4{x}^{2}+4)dx \right]\end{equation*}
    
    From here just simplify and use the power rule.
    
    \begin{figure}[h]
    	\centering
    		\includegraphics[width=0.90\textwidth]{plot.jpg}
    \end{figure}
    
    
    \end{document}
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  3. #3
    MHF Contributor
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    Edit: didn't see earlier post
    y=x^4 -4x^2
    which factorises to
    y=x^2(x - 2)(x + 2)


    Find the intersection points:
    x^2(x - 2)(x + 2)=x^2

    [first solution: x=0]

    (x - 2)(x + 2)=1


    this is a quadratic. solve to obtain x=+/- sqrt{5}

    http://www.wolframalpha.com/input/?i...x^2%2C+y%3Dx^2


    This means the limits for your integration should be:
    (-sqrt{5}, 0)

    (0, sqrt{5})

    will that solve your problem?
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  4. #4
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    Thumbs up Gracias!

    Thank you very much TheEmptySet. I've found the answer which is 8. Hopefully I'm right. I'm gonna make a scan and post it:



    I apologize for the smudges you see. Those are from the previous page (I write very deep) =/
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