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Easy Polar Coordinates question (Change of variables)

  1. Oct 20, 2011 #1
    I have a question regarding problem solving tips.

    When given an iterated integral and asked to convert it to polar coordinates, how do you select the bounds of theta - do you have to understand how the graph of r operates and therefore know where the theta bounds are based on the rectangular coordinate bounds (aka completely conceptual)? Or is there a mathematical way of solving for the theta bounds?

    For instance:
    Set up a double integral in polar coordinates for:
    f(x,y) = x+y
    R: x^2 + y^2 ≤ 4
    x ≥ 0
    y ≥ 0

    Obviously, the theta bounds are from 0->pi/2 because of the y and x bounds, but is there a mathematical procedure to arrive at this same answer? Or must you figure it out conceptually.

    Thank you in advance for your help!
     
    Last edited: Oct 20, 2011
  2. jcsd
  3. Oct 20, 2011 #2

    LCKurtz

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    You use the graph, perhaps with a little algebra to find intersection points. Just like in rectangular coordinates. You always (should!) draw the graph first.
     
  4. Oct 21, 2011 #3

    HallsofIvy

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    Since [itex]x= r cos(\theta)[/itex] the boundary line x= 0 corresponds to [itex]\theta= 0[/itex]. Since [itex]y= r sin(\theta)[/itex], the boundary line y= 0 corresponds to [itex]\theta = \pi/2[/itex]. Since [itex]x^2+ y^2= r^2[/itex], the boundary [itex]x^2+ y^2= 1[/itex] gives [itex]r^2= 1[/itex] or r= 1 since r cannot be negative.
     
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