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Thread: Integration of absolute value function.. please help

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    Integration of absolute value function.. please help

    Integration of absolute value function.. please help-question-1.jpg
    Let f(a) = S02|x(x-a)| dx for 0< a <

    (1) Find the function f(a).

    (2) Find the minimum of f(a).


    Thank you very much for your help!
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    Re: Integration of absolute value function.. please help

    Quote Originally Posted by kokoro123 View Post
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    Let f(a) = S02|x(x-a)| dx for 0< a <

    (1) Find the function f(a).

    (2) Find the minimum of f(a).
    The integral becomes $\displaystyle \int_0^a {x(a - x)dx} + \int_a^2 {x(x - a)dx}$
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    Re: Integration of absolute value function.. please help

    Quote Originally Posted by kokoro123 View Post
    (1) Find the function f(a).
    Split it up:

    $\displaystyle \int_0^2\left|x(x-a)\right|\,dx$

    $\displaystyle =\int_0^a\left|x(x-a)\right|\,dx + \int_a^2\left|x(x-a)\right|\,dx$

    $\displaystyle =\int_0^a\left[-x(x-a)\right]\,dx + \int_a^2x(x-a)\,dx$

    Quote Originally Posted by kokoro123 View Post
    (2) Find the minimum of f(a).
    Use the Fundamental Theorem of Calculus to differentiate $\displaystyle f$. Locate critical values, etc.
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