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Thread: 2 dimensional integral

  1. #1
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    2 dimensional integral

    Can some one give me an example of a 2 dimensional integral over a region Q, such that one of the iterated integrals exist, but the other does not.

    I have tried sort of taking examples of functions that are not continuous, but the ones I have set up, just don't fit into the description.
    I appreciate any help I can get.
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  2. #2
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    Re: 2 dimensional integral

    Here's a simple example: Find the area between the graph of y= 1/x and y= 0, from x= 1 to infinity.

    That would be [LaTeX ERROR: Compile failed] . The first integral, from 0 to 1/x, with respect to y, exists and is 1/x. The second integral, then, is [LaTeX ERROR: Compile failed] which does not exist.

    Or do you mean [LaTeX ERROR: Compile failed] , say, exists, but [LaTeX ERROR: Compile failed] does not? In that case, if one exists, the other must and must give the same result.
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  3. #3
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    Re: 2 dimensional integral

    Quote Originally Posted by HallsofIvy View Post
    Here's a simple example: Find the area between the graph of y= 1/x and y= 0, from x= 1 to infinity.

    That would be $\int_1^\infty \int_0^{1/x} dydx$. The first integral, from 0 to 1/x, with respect to y, exists and is 1/x. The second integral, then, is $\int_1^\infty \frac{1}{x} dx= \left[ln(x)\right]_1^\infty$ which does not exist.

    Or do you mean $\int\int f(x,y)dxdy$, say, exists, but $\int\int f(x,y)dydx$ does not? In that case, if one exists, the other must and must give the same result.
    repost to fix latex
    Thanks from skeeter and HallsofIvy
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