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Finding limits of integral in spherical coordinates

  1. Apr 25, 2015 #1
    1. The problem statement, all variables and given/known data
    The question asks me to convert the following integral to spherical coordinates and to solve it
    ?temp_hash=ddb394627c22cb4cf1570d8159d5312b.jpg

    2. Relevant equations


    3. The attempt at a solution
    just the notations θ = theta and ∅= phi

    dx dy dz = r2 sinθ dr dθ d∅
    r2 sinθ being the jacobian

    and eventually solving gets me
    ∫ ∫ ∫ r4 *sin2θ * sin∅ dr dθ d∅

    How do I find the limits now?
     

    Attached Files:

  2. jcsd
  3. Apr 25, 2015 #2

    ehild

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    Use the limits in the Cartesian system to figure out the enclosed shape. What is the minimum and maximum value of z? Those of y and x?
     
  4. Apr 25, 2015 #3
    x goes from -2 to 2
    y goes from 0 to √4-x2 circle with radius 2
    z from 0 to √4-x2-y2 sphere with with radius 2

    so Im guessing
    r goes from 0 to 2
    ∅ and θ from 0 to 2π
     
  5. Apr 25, 2015 #4

    ehild

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    Are you sure in 2pi?
     
  6. Apr 25, 2015 #5
    ok so ∅ goes from 0 to 2pi as it is some sort of sphere/ elipse. correct?
     
  7. Apr 25, 2015 #6

    ehild

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    See picture. Yes, the shape is spherical, but you have to integrate with respect to y from zero to some positive value, goes it round a whole circle?

    intshape.JPG
     
  8. Apr 25, 2015 #7
    ahhh so θ goes from 0 to pi
    ∅ goes from 0 to 2pi
    and r goes from 0 to 2
    correct?
     
  9. Apr 25, 2015 #8

    ehild

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    Those were the limit for the whole sphere. But the integration does not go for negative z values, neither for negative y values.
     
  10. Apr 26, 2015 #9
    r goes from 0 to 2
    theta goes from 0 to pi/2
    phi goes from 0 to pi
    correct?
     
    Last edited: Apr 26, 2015
  11. Apr 26, 2015 #10

    BvU

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    Looks good.
     
  12. Apr 26, 2015 #11

    ehild

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    Yes. :smile:
     
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