evaluate the integral

• Jan 8th 2011, 09:42 AM
wopashui
evaluate the integral
Let C be the semi circle on the surface $\displaystyle x^2+y^2+z^2=4$ from $\displaystyle N=(0,0,2)$to $\displaystyle S=(0,0,-2)$ which passes through the point ($\displaystyle \sqrt{2}$,$\displaystyle \sqrt{2}$,$\displaystyle 0$) (note that x=y for all (x,y,z) on C.)
Evaluate the integral
$\displaystyle \int _c\!z^2dx + 2x^2dy + xydz$

(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?
• Jan 8th 2011, 08:15 PM
snowtea
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
$\displaystyle 4 = x^2 + y^2 + z^2 = 2x^2 + z^2 = 2y^2 + z^2$
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?