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Thread: integration with boundary conditions

  1. #1
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    integration with boundary conditions

    Suppose we have a function f(x) defined on the interval [x_1,x_2] whose antiderivative is F(x) + C.

    If F(x_1) = K_1 and F(x_2) = K_2 then how does one determine the value of C?

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  2. #2
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    Re: integration with boundary conditions

    The way you have given it here, you can't determine C. But I suspect that is not what you mean to say. If the anti-derivative of f(x) is F(x)+ C, then you already know F(x) so already know both F(x_1) and F(x_2) and that tells you nothing about C. For example, I know that the anti-derivative of f(x)= 2x is \int f(x) dx= F(x)+ C= x^2+ C so I know that F(x_1)= x_1^2 and F(x_2)= x_2^2. That tells me nothing about C.

    I think what you mean to say is that you know that \int f(x)dx, evaluated at x_1, is K_1 (and you only need one value, not two). In that case, you have that F(x_1)+ C= K so that C= K- F(x_1). Using the example above, if you know that \int x^2 dx, evaluated at x_1 is K then x_1^2+ C= K_1 so C= K_1- x_1^2. More specifically, if you know that the anti-derivative of f(x)= 2x, evaluated at x= 2 is 12, then you know that (2)^2+ C= 4+ C= 12 so that C= 8.
    Thanks from jumbo1985
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