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Math Help - Find the area

  1. #1
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    Find the area

    y=8x2x3+x
    y=x2+11x



    Find The Total area enclosed by the graphs of these 2 functions.

    Can anyone show me how to solve this without graphing it, we need to make it into an integral and I can't seem to figure out the answer, any help would be much appreciated.
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  2. #2
    Super Member Showcase_22's Avatar
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    I might be wrong, but I think you can do it like this:

    8x^2-x^3+x=x^2+11x

    7x^2-x^3-10x=0

    x^2-7x+10=0

    (x-2)(x-5)=0

    x=2  \ or \ 5.

    So we want \int_{2}^{5} x^2-7x+10dx.
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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    Quote Originally Posted by Showcase_22 View Post

    7x^2-x^3-10x=0

    x^2-7x+10=0
    Don't ever do that again.

    Your work was for nice way, so we have to split the given area into two integrals:

    \int_{0}^{2}{\alpha (x)\,dx}+\int_{2}^{5}{\beta (x)\,dx}.
    Where \alpha,\beta are some functions to being integrated.

    Now, put f(x)=8x^2-x^3+x and g(x)=x^2+11x. Note that \alpha(x)=g(x)-f(x)\ge0 on [0,2] and \beta(x)=f(x)-g(x)\ge0 on [2,5]. Finally, the required area is \mathcal A=\int_{0}^{2}{\big(g(x)-f(x)\big)\,dx}+\int_{2}^{5}{\big(f(x)-g(x)\big)\,dx}.
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