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Math Help - Evaluate the Integral

  1. #1
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    Evaluate the Integral

    \int_0^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} (x^2+y^2) dydx

    I dont know how to change it to cylindrical coordinates and set up the integral for it.
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  2. #2
    Senior Member yeKciM's Avatar
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    u can use here polar coordinates

     x= \rho cos \varphi

     y= \rho sin \varphi

     J= \rho

    so it will be

    \displaystyle \int_0 ^3 \int _{-\sqrt{9-x^2}} ^{\sqrt{9-x^2}} \rho^2 \rho \; d\varphi\; d\rho

    and u just need to change limits simple


    P.S here u are cylindrical coordinates

     x= \rho cos \varphi

    y= \rho sin \varphi

    z=z

     J= \rho



    lol i think u need this

     \displaystyle \int_{-\frac {\pi}{2}} ^{\frac {\pi}{2}} d\varphi \int _0 ^3 \rho^3 \; d\rho
    Last edited by yeKciM; July 30th 2010 at 11:51 AM.
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  3. #3
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    First draw a picture. Set up an xy- coordinate system. Since x ranges between x= 0 and x= 3, draw vertical lines there. Now, for each x, y ranges between y= -\sqrt{9- x^2} and y= \sqrt{9- x^2}. But squaring either of those gives y^2= 9- x^2 or x^2+ y^2= 9, the circle with center at (0,0) and radius 3. The right half of that circle lies between x= 0 and x= 3.

    That is, the region you are integrating is a semi-circle, the portion of x^2+ y^2= 9 in the x> 0 half-plane.

    Of course, x^2+ y^2= r^2 and dxdy= r drd\theta.

    That gives the integral yeKciM gives.
    (Except that I would use "r" rather than " \rho". I reserve " \rho" for spherical coordinates.)
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  4. #4
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    First draw a picture. Set up an xy- coordinate system. Since x ranges between x= 0 and x= 3, draw vertical lines there. Now, for each x, y ranges between y= -\sqrt{9- x^2} and y= \sqrt{9- x^2}. But squaring either of those gives y^2= 9- x^2 or x^2+ y^2= 9, the circle with center at (0,0) and radius 3. The right half of that circle lies between x= 0 and x= 3.

    That is, the region you are integrating is a semi-circle, the portion of x^2+ y^2= 9 in the x> 0 half-plane.

    Of course, x^2+ y^2= r^2 and dxdy= r drd\theta.

    That gives the integral yeKciM gives.
    (Except that I would use "r" rather than " \rho". I reserve " \rho" for spherical coordinates.)
    and looks like this when drown



    yes, you are correct but it's (for me at least) much easier to remember (or don't forget) when using like this

    Polar coordinates :  (x,y) \to (\rho , \varphi)

     x = \rho \cos \varphi

     y = \rho \sin \varphi

     J= \rho

    natural limits for polar coordinates :

     \rho \ge 0

    \displaystyle \varphi \mid _0 ^{2\pi}


    Cylindrical coordinates :  (x,y,z) \to (\rho , \varphi,z)

     x = \rho \cos \varphi

     y = \rho \sin \varphi

     z = z

     J= \rho

    natural limits for cylindrical coordinates :

     \rho \ge 0

    \displaystyle \varphi \mid _0 ^{2\pi}

    \displaytyle z \mid _{-\infty} ^{\infty}


    Spherical coordinates :  (x,y,z) \to (\rho ,\varphi, \theta)

     x= \rho \cos( \varphi ) \sin (\theta)

     y= \rho \sin( \varphi ) \sin (\theta)

     z= \rho  \cos (\theta)

     J= \rho^2 \sin (\theta)

    natural limits for spherical coordinates:

     \rho \ge 0

     \varphi \mid _0 ^{2\pi}

     \theta \mid _0 ^\pi



    P.S. @ larryboi7 : when changing the limits u must always know that they can't be greater than the natural limits , but can be smaller ... let's say for spherical \theta goes from 0 to \pi, and never can go to let's say from 0 to \frac {3\pi}{2} but can from 0 to \frac {\pi}{2} just for example

    and if not by any condition in your task limits of \rho, \varphi, \theta are changed u'll use natural limits for those



    and what is more important (if u don't realize it) when for ur integral changing limits for \rho it's obvious that limits are from 0 to 3, because u have circle with center in (0,0) so ur limits goes from zero to radius of ur circle, but if ur circle is shifted (doesn't have center in point (0,0) ) then u must do a little calculating to see how do your \rho changes...
    Last edited by yeKciM; July 31st 2010 at 06:23 AM.
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