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Integrals in cylindrical coordinates.

  1. Nov 4, 2013 #1
    Integrate the function f(x,y,z)=−7x+2y over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and y=sqrt(263/137)x and contained in a sphere centered at the origin with radius 25 and a cone opening upwards from the origin with top radius 20.

    I am not sure I am getting the right picture. Here are the bounds for the integral I found.

    arctan(sqrt(263/137)) <= theta <= pi/2
    0 <= z < 15
    0 <= r <= (4/3)z

    I am integrating in the order of dr dz d_theta.

    The integrand I cam up with is...
    -7r2 * cos(theta) + 2r2 * sin(theta) dr dz d_theta

    Can anyone tell me where I went wrong. I keep getting a crazy answer.
     
  2. jcsd
  3. Nov 4, 2013 #2

    ehild

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    You did it well, the bounds are correct. What is the result?

    ehild
     
  4. Nov 4, 2013 #3
    I finally did the integral right and I get 3500 * sqrt(263) + 1000 * sqrt(137) - 70000 which is about -1534.83. I put the exact answer into my web HW and it is wrong. I checked the integral with my CAS calculator and this is what it gets as well. Maybe the answer is positive but I only have six attempts and I don;t want to waste them. Can someone do the integral and tell me what they get?
     
  5. Nov 5, 2013 #4

    ehild

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    Try to integrate from x=0 to the section of the line in the third quadrant, that is from theta=pi/2 to theta = pi+arctan(sqrt(263/137))

    ehild
     
  6. Nov 5, 2013 #5
    I think the problem said it is only in the volume in the first octant.
     
  7. Nov 5, 2013 #6

    ehild

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    I see. Then your result must be correct, but use less significant digits. I would omit the decimals.

    ehild
     
  8. Nov 5, 2013 #7
    With this web HW system I can actually enter the exact value "3500 * sqrt(263) + 1000 * sqrt(137) - 70000".
     
  9. Nov 5, 2013 #8

    ehild

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    Maybe it would do...


    ehild
     
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