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Thread: Combining of 2 Double Integrals into 1: how to convert to polar

  1. #1

    Question Combining of 2 Double Integrals into 1: how to convert to polar

    Hello. On this problem shown in the attached image, I have to combine two double integral sums into one double integral.



    15. This sum of two double integrals may be written as one double integral.

    . . . . .[tex]\color{blue}{ \displaystyle \int_0^3\, \int_0^{\frac{4x}{3}}\, f(x,\, y)\, dy\, dx\, +\, \int_3^5\, \int_0^{\sqrt{\strut 25\, -\, x^2\,}}\, f(x,\, y)\, dy\, dx }[/tex]

    What is this one double integral?

    Answer:


    . . . . .[tex]\color{green}{ \displaystyle \int_0^4\, \int_{\frac{3y}{4}}^{\sqrt{\strut 25\, -\, y^2\,}}\, f(x,\, y)\, dx\, dy }[/tex]

    ...or:

    . . . . .[tex]\color{green}{ \displaystyle \int_0^{0.927}\, \int_0^5\, f(r,\, \theta)\, r\, dr\, d\theta }[/tex]



    Looking at the answer, it seems one easy way to do it is to convert to polar coordinates. How would I go about doing this?
    Attached Images Attached Images
    Last edited by stapel; 12-04-2017 at 09:43 AM. Reason: Typing out the text in the graphic; creating useful subject line.

  2. #2
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    Quote Originally Posted by TheNerdyGinger View Post
    Hello. On this problem shown in the attached image, I have to combine two double integral sums into one double integral.



    15. This sum of two double integrals may be written as one double integral.

    . . . . .[tex]\color{blue}{ \displaystyle \int_0^3\, \int_0^{\frac{4x}{3}}\, f(x,\, y)\, dy\, dx\, +\, \int_3^5\, \int_0^{\sqrt{\strut 25\, -\, x^2\,}}\, f(x,\, y)\, dy\, dx }[/tex]

    What is this one double integral?

    Answer:


    . . . . .[tex]\color{green}{ \displaystyle \int_0^4\, \int_{\frac{3y}{4}}^{\sqrt{\strut 25\, -\, y^2\,}}\, f(x,\, y)\, dx\, dy }[/tex]

    ...or:

    . . . . .[tex]\color{green}{ \displaystyle \int_0^{0.927}\, \int_0^5\, f(r,\, \theta)\, r\, dr\, d\theta }[/tex]



    Looking at the answer, it seems one easy way to do it is to convert to polar coordinates. How would I go about doing this?
    Sketch The Areas of the two double integrals first.
    Last edited by stapel; 12-04-2017 at 09:44 AM. Reason: Copying typed-out graphical content into reply.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3

    Combining of Double Integrals?

    Quote Originally Posted by Subhotosh Khan View Post


    Sketch The Areas of the two double integrals first.
    I drew it and saw that it was a circle with a radius of 5. (Hence r going from 0-5). Also, y = 4pi/3 intersects this circle at x = 2.73. I tried doing the inverse tangent of y/x and got a theta value of 0.993. This isn't quite the answer that was given. What am I missing? Is there something I need to subtract?

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