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Definite integral with variable limits of a multivariable function.

  1. Mar 17, 2013 #1
    I have the following integral:

    [itex]\int_0^{f(x,y)}{f' \sin(y-f')df'}[/itex]

    Now suppose that f(x,y) = x*y, my question is how do I write the integral in terms of x and y only? Can I do something like this?

    Since [itex]df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy[/itex] we can obtain:

    [itex]\int_0^{x y}{x' y' \sin(y'-x' y') (\frac{\partial f'}{\partial x'}dx'+\frac{\partial f'}{\partial y'}dy'})[/itex]
     
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  3. Mar 18, 2013 #2

    HallsofIvy

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    First, you should be careful to use different letters for variables in the integrand and in the limits of integration. So write the integral as
    [tex]\int_0^{f(x,y)} f'(u,v) sin(v- f'(u,v)) d(f'(uv))[/tex]

    Now, what do you mean by f' where f is a function of two variables? The derivative with respect to some parameter, t, so that [itex]df= (\partial f)(\partial x)(dx/dt)+ (\partial f)(\partial y)(dy/dt)[/itex]? (NOT df'- that would involve two derivatives).

    First, of course, if f= xy, then f' is NOT x'y'. By the product rule, f'= xy'+ x'y. And "y" in the sine would not automatically become y'. The integral would be
    [tex]\int_0^{x(t)y(t)}(x(s)y'(s)+ x'(s)y(s))sin(y(s)+ x'(s)y(s)+ x(s)y'(s))(f_x(x(s),y(s)) x'(s)+ f_y(x(s),y(s))y'(s)) ds[/tex]
     
  4. Mar 18, 2013 #3

    Thanks for your reply, I apologize for the confusion but I don't think I was clear, the apostrophe was meant to indicate different letters for variables, not a derivative. With that in mind, I will rephrase my question and actually simplify the integrand since that is not what is important.

    Suppose we have [tex]\int^{f(x,y)}_0{g(u,v)dg}[/tex]. Given [tex]f(x,y) = xy[/tex] and [tex]g(u,v) = uv[/tex], now since [tex]dg=\frac{\partial g}{\partial u}du+\frac{\partial g}{\partial v}dv = vdu + udv[/tex], is the integral then equal to:


    [tex]\int^{f(x,y)}_0{g(u,v)dg} = \int^{xy}_0{u v^2 du} + \int^{xy}_0{u^2 v dv}[/tex]
     
    Last edited: Mar 18, 2013
  5. Mar 18, 2013 #4

    HallsofIvy

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    What you have written makes no sense. You can certainly have [tex]\int g dg[/tex] but "g" now is a dummy variable- you cannot then assert that g= uv.

    What is true is that [tex]\int_0^{xy} g dg= \left[\frac{1}{2}g^2\right]_0^{xy}= \frac{1}{2}x^2y^2[/tex].
     
  6. Mar 18, 2013 #5
    Excellent. I knew it didn't make sense and just needed confirmation, I got the integral from an obscure paper on a computational code and that confirms what I suspected - the integral is not written properly. Thanks.
     
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