evaluate the integral

**wopashui** · January 8th 2011, 09:42 AM

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?

**snowtea** · January 8th 2011, 08:15 PM

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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