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Thread: evaluate the integral

  1. #1
    Senior Member
    Jan 2010

    evaluate the integral

    Let C be the semi circle on the surface x^2+y^2+z^2=4 from N=(0,0,2) to S=(0,0,-2) which passes through the point ( \sqrt{2}, \sqrt{2}, 0) (note that x=y for all (x,y,z) on C.)
    Evaluate the integral
    \int _c\!z^2dx + 2x^2dy + xydz

    (Suggestion: use as your parameter the angle Q subtended at the orgin by the arc NP for a point P on C)

    I think i need to sketch the picture to understand it, but Im having a hard time to do it, is there another way to find x,y,z in term of t?
    Last edited by wopashui; Jan 8th 2011 at 07:26 PM.
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  2. #2
    Senior Member
    Dec 2010
    Your semi circle is part of the intersection between the plane x=y and the sphere of radius 2.

    So points on your semicircle satisfy
    4 = x^2 + y^2 + z^2 = 2x^2 + z^2 = 2y^2 + z^2
    Can you find parametric equations for x, y, and z on the semicircle?

    Hint: x = y = Asin(t), z = Bcos(t), what are A,B, and the range for t?
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