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Finding the volume under a plane and region (polar coordinates)

  1. Dec 1, 2006 #1
    Hey im trying to compute the volume of the region under the plane z=7 x + 4 y + 34 and over the region in the xy -plane bounded by the circle x^2+y^2=4 y.

    i cant seem to get it.... like i i know that the circle is x^2+(x-2)^2=4
    so 0<r<2 and 0<theta<2pi

    this is what i try
    double integral of (7x+4y+34) in polar tho... and it doesnt work... wat am i doinig wrong.. and i cant seem to center the circle... any help would be much appreciated...
     
  2. jcsd
  3. Dec 1, 2006 #2

    HallsofIvy

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    You mean, of course, x2+ (y-2)2= 4. More importantly, that region in the plane is NOT given by 0< r< 2, 0< [itex]\theta[/itex]< [itex]2\pi[/itex]. That describes a circle of radius 2 centered at (0,0)

    Change coordinates. Let x'= x, y'= y- 2 so that (0, 2), the center of the circle in xy coordinates, becomes (0,0) in x'y' coordinates. Since x= x', y= y'+2, The equation of the circle is now x'2+ y'2= 4. Replace x, y in the equation of the plane by x'= x, y'= y+ 2 and integrate.
     
  4. Dec 1, 2006 #3
    thx alot HallsofIvy, much appreciated!
     
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