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Math Help - Calc 3- Evaluating a triple integral using cylindrical coordinates

  1. #1
    Junior Member krtica's Avatar
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    Calc 3- Evaluating a triple integral using cylindrical coordinates

    Use cylindrical coordinates to evaluate the triple integral , where is the solid bounded by the circular paraboloid and the xy-plane.


    I'm having trouble deciding what the bounds for r would be.
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  2. #2
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by krtica View Post
    Use cylindrical coordinates to evaluate the triple integral , where is the solid bounded by the circular paraboloid and the xy-plane.


    I'm having trouble deciding what the bounds for r would be.
    Let z = 0 to find the domain in the xy plane

     0 = 9 -16 (x^2 + y^2)

     9/16 = x^2 + y^2

    When we transform to polar this becomes

     \sqrt{9/16} = r

    Therefore

     0 \le r \le 3/4

    If you recall that in polar

     dA = rdr d \theta dz

    Then we have

     V= \int_0^{2 \pi} d \theta \int_0^{ 3/4} r^2dr \int_0^{9-16r^2} dz
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