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Average value of f(x,y) = xy in quarter circle x^2 + y^2 < 1 in Q1.

  1. Apr 13, 2005 #1
    Here is the problem:

    Find the average value of [tex]f\left(x, y\right) = x\;y[/tex] for the quarter circle [tex]x^2 + y^2 \le 1[/tex] in the first quadrant.

    Here is what I have:

    Average value equation is [tex]\frac{1}{Area\;of\;R} \iint_{R} f\left(x, y\right) dA[/tex]

    [tex]f\left(x, y\right) = x\;y = \left(r\;\cos\theta\right)\left(r\;\sin\theta\right)[/tex]

    The area of one quarter of a unti circle is [tex]\frac{\pi}{4}[/tex], right?

    [tex]Average = \frac{4}{\pi}\;\int_{0}^{\frac{\pi}{2}}\int_{0}^{1}\;\left(r\;\cos\theta\right)\left(r\;\sin\theta\right)\;r\;dr\;d\theta = \frac{1}{2\pi}[/tex]

    Is this correct?
  2. jcsd
  3. Apr 13, 2005 #2
    Absolutely, and good job remembering the jacobian in polar coordinates.
  4. Apr 13, 2005 #3
    Thank you

    Many thanks. I will probably be checking a few others before I am through.

    I wonder if I should make a thread for "Check my Calculus Answers Please" and put them all in there or something? On the other hand, I think that individually posting each problem gets quicker results and is easier to search (i guess).
  5. Apr 13, 2005 #4
    In order to get useful feedback and responses, it's best to title your threads appropriately--- putting something like "check my work..." generally isn't too appealing, so you're probably right--- that is, make threads that are individual for a specific problem, with the statement and reasoning placed in the first post.

    You have been doing this but just so others know!
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