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Math Help - volume

  1. #1
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    volume

    express as a repeated integral the volume of the cone bounded above by z=2 and below by z= (x^2 +Y^2) ^(1/2) by taking slices at fixed x.


    i was trying to find the limits for x and y and z but i got stuck..

    since im taking slices at fixed x, -2<x<2

    and (x^2 +Y^2) ^(1/2) <z<2

    but i do not know how to find the limits for y.

    thanks
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    express as a repeated integral the volume of the cone bounded above by z=2 and below by z= (x^2 +Y^2) ^(1/2) by taking slices at fixed x.


    i was trying to find the limits for x and y and z but i got stuck..

    since im taking slices at fixed x, -2<x<2

    and (x^2 +Y^2) ^(1/2) <z<2

    but i do not know how to find the limits for y.

    thanks
    Ok, the limits +/-2 for the outermost integral with respect to x are right, now just imagine that an x from that range has been chosen. The y over which the next inner integral varies must still satisfy \sqrt{x^2+y^2}\leq z\leq 2.

    Since z can still be chosen to be from a suitably small intervall, the relevant condition for the limits for y is \sqrt{x^2+y^2}\leq 2, from which it follows that y^2\leq 4-x^2, in other words -\sqrt{4-x^2}\leq y\leq +\sqrt{4-x^2}, agreed?
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  3. #3
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    thanks for the explanation...

    because my notes wrote that the range of y should be

    -(2-x^2) ^(1/2) < y< (2-x^2) ^(1/2) ...and i couldnt figure out why...

    i think there should be something wrong with it..
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  4. #4
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by Failure View Post
    Ok, the limits +/-2 for the outermost integral with respect to x are right, now just imagine that an x from that range has been chosen. The y over which the next inner integral varies must still satisfy \sqrt{x^2+y^2}\leq z\leq 2.

    Since z can still be chosen to be from a suitably small intervall, the relevant condition for the limits for y is \sqrt{x^2+y^2}\leq 2, from which it follows that y^2\leq 4-x^2, in other words -\sqrt{4-x^2}\leq y\leq +\sqrt{4-x^2}, agreed?
    I agree with this

    Quote Originally Posted by alexandrabel90 View Post
    thanks for the explanation...

    because my notes wrote that the range of y should be

    -(2-x^2) ^(1/2) < y< (2-x^2) ^(1/2) ...and i couldnt figure out why...

    i think there should be something wrong with it..
    I think your prof must have a mistake. Seeing as this is a cylindrical cone would we not want to do this in cylindrical co-ordinates anyways?
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  5. #5
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    thank you! (:
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