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Math Help - Polar Co-ordinates

  1. #1
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    Polar Co-ordinates

    Find the surface area of the circular paraboloid
    z = x^2+ y^2" alt="z = x^2+ y^2" />between

    the
    xy-plane and the plane z = 4.

    The limits have suddeny become for double integration

    2pi 0
    and 3 and 0
    dr dtheta

    I know

    r = (x^2 + y^2)^0.5
    x = r cos theta
    y = r sin theta
    and theta = inverse tan (x/y)

    How do i get these limits? Thanks.
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  2. #2
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    z = x^2 + y^ 2 is the error
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    The surface area is

    S=\displaystyle\iint_{x^2+y^2\leq 4}\sqrt{1+4x^2+4y^2}\;dxdy=

    \displaystyle\int_{0}^{2\pi}d\theta\displaystyle\i  nt_{0}^{2}\sqrt{1+4r^2}\;r\;dr=\ldots
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  4. #4
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    Quote Originally Posted by FernandoRevilla View Post
    The surface area is

    S=\displaystyle\iint_{x^2+y^2\leq 4}\sqrt{1+4x^2+4y^2}\;dxdy=

    \displaystyle\int_{0}^{2\pi}d\theta\displaystyle\i  nt_{0}^{2}\sqrt{1+4r^2}\;r\;dr=\ldots
    According to my book its 3 and 0.

    How did you calculate the limits, i don't understand?
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by adam_leeds View Post
    According to my book its 3 and 0.

    What is 3 and 0?. Could you rewrite the question?. Your first post says the plane z=4, so the projection over the xy plane is x^2+y^4\leq 4 .
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  6. #6
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    Quote Originally Posted by FernandoRevilla View Post
    What is 3 and 0?. Could you rewrite the question?. Your first post says the plane z=4, so the projection over the xy plane is x^2+y^4\leq 4 .
    Your 2nd integral you have got between 2 and 0 my book says its between 3 and 0
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  7. #7
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    Can you just tell me how to work it out please? Thanks.
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by adam_leeds View Post
    Your 2nd integral you have got between 2 and 0 my book says its between 3 and 0

    Your book is wrong.
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  9. #9
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by adam_leeds View Post
    Can you just tell me how to work it out please? Thanks.
    Better fifty fifty.

    First question:

    Could you please write the projection of the given surface over the xy plane ?
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  10. #10
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    Quote Originally Posted by FernandoRevilla View Post
    Better fifty fifty.

    First question:

    Could you please write the projection of the given surface over the xy plane ?
    It just says

    Find the surface area of the circular paraboloid z = x^2 + y^2 between the xy-plane and the plane z= 4
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  11. #11
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by adam_leeds View Post
    It just says. Find the surface area of the circular paraboloid z = x^2 + y^2 between the xy-plane and the plane z= 4

    Well, look at the answer #3 and tell me exactly what you don't undesrtand.
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