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Double integral with polar coordinates

  1. May 14, 2012 #1
    1. The problem statement, all variables and given/known data
    It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ.

    f(x,y)=1 (plane parallel to Oxy plane)

    They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it.


    2. Relevant equations

    x=rcosθ
    y=rsenθ
    r=√x2+y2

    3. The attempt at a solution

    I've done the variable substitution as:

    0≤rcosθ≤1, 0≤rsenθ≤1-cosθ and ∫∫Setrdrdθ

    After analysing it for a bit I figured that 0≤r≤1 and that 0≤θ≤[itex]\frac{\pi}{2}[/itex].
    However, the solution to the integral is 0.5. For the limits I've established, it gives me [itex]\frac{\pi}{4}[/itex].

    I can easily calculate that integral in x,y coordinates but I'm having trouble defining the endpoints of r and θ when changing a set from x,y coordinates to r and θ coordinates.

    Can you help me with this?
     
  2. jcsd
  3. May 14, 2012 #2

    LCKurtz

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    ##r## doesn't go from 0 to 1. In your picture, pick some ##\theta## and draw the ##r## for that ##\theta##. ##r## goes from 0 to the ##r## value on the line. So write the equation of the line in polar coordinates and solve it for ##r##. That is your upper limit on ##r##.
     
  4. May 14, 2012 #3

    sharks

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    What's senθ? :smile:

    Using your given limits for x and y, you should draw the graph, so you can understand and derive the limits for polar coordinates.
     
    Last edited: May 14, 2012
  5. May 14, 2012 #4
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