- #1

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## Homework Statement

Let C be the semi-circle on the sphere [itex]x^2+y^2+z^2 = 2[/itex] from [itex]N = (0,0,\sqrt{2})[/itex] to [itex]S = (0,0, - \sqrt{2})[/itex] which passes through the point [itex](1,1,0)[/itex]

Note that x=y for all (x,y,z) on C. Evaluate the integral :

[itex]\int_C z^2dx + 2x^2dy +xydz[/itex]

Hint : Use as your parameter the angle [itex]θ[/itex] subtended at the origin by the arc NP for a point P on C.

## Homework Equations

N/A

## The Attempt at a Solution

So I wasn't sure how to get this one going. I'm told that C is a semi-circle on the sphere [itex]x^2+y^2+z^2 = 2[/itex] from one endpoint N to the other endpoint S which passes through (1,1,0).

So I know my first step is to parametrize using the angle θ.

So : [itex]x = cosθ, y = sinθ, z = ?[/itex] I'm thinking that z = θ. As for the interval of θ, I'm not quite sure.

Once I set the integral up, it will be easy to evaluate. I've never had a case of 3 variables over anything but lines so I'm a bit confused. I'm also thinking I may have to split this integral.

Thanks for any help in advance.