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Thread: integration with boundary conditions

  1. #1
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    integration with boundary conditions

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

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

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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 $\displaystyle F(x_1)$ and $\displaystyle F(x_2)$ and that tells you nothing about C. For example, I know that the anti-derivative of $\displaystyle f(x)= 2x$ is $\displaystyle \int f(x) dx= F(x)+ C= x^2+ C$ so I know that $\displaystyle F(x_1)= x_1^2$ and $\displaystyle F(x_2)= x_2^2$. That tells me nothing about C.

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