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Math Help - Reverse the order of integration

  1. #1
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    Reverse the order of integration

    How do you do reverse order of integration for an iterated integral?
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    Quote Originally Posted by ottmar0 View Post
    How do you do reverse order of integration for an iterated integral?
    You've asked what's called a fat question.

    Can you give us the question you're working on so that we can guide you through it?
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    Here is the problem

    double integral of the function of where y is greater than equal to tanpie/4 *x or less than or equal to x, and x is greater than 0 and less than or equal to 1.
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    Quote Originally Posted by ottmar0 View Post
    double integral of the function of where y is greater than equal to tanpie/4 *x or less than or equal to x, and x is greater than 0 and less than or equal to 1.
    Is this right?

    \int_{\tan{\left(\frac{\pi x}{4}\right)}}^x{\int_0^1{f(x, y)\,dx}\,dy}
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    change

    switch them around, 1 with the x, and 0 with tan(piex/4) dydx.
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    Quote Originally Posted by ottmar0 View Post
    switch them around, 1 with the x, and 0 with tan(piex/4) dydx.
    That's not what you asked for, but ok...

    \int_0^1{\int_{\tan{\left(\frac{\pi x}{4}\right)}}^x{f(x, y)\,dy}\,dx}


    How's this?
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    right.

    thats it
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    Quote Originally Posted by Prove It View Post
    Is this right?

    \int_{\tan{\left(\frac{\pi x}{4}\right)}}^x{\int_0^1{f(x, y)\,dx}\,dy}
    I find the easiest way to reverse orders of integration is to look at the inequalities and "solve" them simultaneously.

    Notice that

    \tan{\left(\frac{\pi x}{4}\right)} \leq y \leq x and 0 \leq x \leq 1.


    You should be able to see that since y \leq x and x \leq 1, then y \leq 1.

    Also, if \tan{\left(\frac{\pi x}{4}\right)} \leq y and 0 \leq x, then 0 \leq y.

    Finally, if \tan{\left(\frac{\pi x}{4}\right)} \leq y, then

    \frac{\pi x}{4} \leq \arctan{y}

    x \leq \frac{4}{\pi}\arctan{y}.


    So your new inequalities are

    0 \leq y \leq 1 and y \leq x \leq \frac{4}{\pi}\arctan{y}.



    Thus the double integral is

    \int_0^1{\int_{\tan{\left(\frac{\pi x}{4}\right)}}^x{f(x, y)\,dy}\,dx} = \int_0^1{\int_y^{\frac{4}{\pi}\arctan{y}}{f(x, y)\,dx}\,dy}.
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    Quote Originally Posted by Prove It View Post
    I find the easiest way to reverse orders of integration is to look at the inequalities and "solve" them simultaneously.

    Notice that

    \tan{\left(\frac{\pi x}{4}\right)} \leq y \leq x and 0 \leq x \leq 1.


    You should be able to see that since y \leq x and x \leq 1, then y \leq 1.

    Also, if \tan{\left(\frac{\pi x}{4}\right)} \leq y and 0 \leq x, then 0 \leq y.

    Finally, if \tan{\left(\frac{\pi x}{4}\right)} \leq y, then

    \frac{\pi x}{4} \leq \arctan{y}

    x \leq \frac{4}{\pi}\arctan{y}.


    So your new inequalities are

    0 \leq y \leq 1 and y \leq x \leq \frac{4}{\pi}\arctan{y}.



    Thus the double integral is

    \int_0^1{\int_{\tan{\left(\frac{\pi x}{4}\right)}}^x{f(x, y)\,dy}\,dx} = \int_0^1{\int_y^{\frac{4}{\pi}\arctan{y}}{f(x, y)\,dx}\,dy}.
    cool~! i've learned it in Calculus III
    the changes of interval can also be explain graphically.. it would be the easiest explaination..
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  10. #10
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    Quote Originally Posted by Prove It View Post
    I find the easiest way to reverse orders of integration is to look at the inequalities and "solve" them simultaneously.

    Notice that

    \tan{\left(\frac{\pi x}{4}\right)} \leq y \leq x and 0 \leq x \leq 1.


    You should be able to see that since y \leq x and x \leq 1, then y \leq 1.

    Also, if \tan{\left(\frac{\pi x}{4}\right)} \leq y and 0 \leq x, then 0 \leq y.

    Finally, if \tan{\left(\frac{\pi x}{4}\right)} \leq y, then

    \frac{\pi x}{4} \leq \arctan{y}

    x \leq \frac{4}{\pi}\arctan{y}.


    So your new inequalities are

    0 \leq y \leq 1 and y \leq x \leq \frac{4}{\pi}\arctan{y}.



    Thus the double integral is

    \int_0^1{\int_{\tan{\left(\frac{\pi x}{4}\right)}}^x{f(x, y)\,dy}\,dx} = \int_0^1{\int_y^{\frac{4}{\pi}\arctan{y}}{f(x, y)\,dx}\,dy}.
    thanks alot, its really helpful. I hate the graph sketching method and you just provided a simpler alternative. cheers
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