Mass of wire on a plane intersecting the unit sphere

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petrol.veem

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I'm working on line integrals right now and came across the following problem:

Find the mass of a wire formed by the intersection of the unit sphere and the plane x + y + z = 0 if the density is given by p(x,y,z)=x^2

This problems seems straight forward enough once i can parametrize the equation, but this is where i'm having problems.

So far, after setting the unit sphere equal to the plane, i have:

x^2 + y^2 + z^2 - 1 = x + y + z

But where to now?
 
petrol.veem said:
I'm working on line integrals right now and came across the following problem:

Find the mass of a wire formed by the intersection of the unit sphere and the plane x + y + z = 0 if the density is given by p(x,y,z)=x^2

This problems seems straight forward enough once i can parametrize the equation, but this is where i'm having problems.

So far, after setting the unit sphere equal to the plane, i have:

x^2 + y^2 + z^2 - 1 = x + y + z

(x - 1/2)^2 + (y - 1/2)^2 + z^2 = 3/2

This is a circle on the plane x + y + z = 0 ...now continue

But where to now?
Click to expand...
 
petrol.veem said:
what happened to the lone z on the right hand side of the equation?
Click to expand...
I suspect the omission is just a typo. :oops:

Make the necessary adjustment, and proceed on your way! :D

Eliz.
 
So completing the square,

(x-1/2)^2 + (y-1/2)^2 + (z-1/2)^2 = 7/4

Now i would like to do something along the lines of

x-1/2 = cost
y-1/2 = sint

but what to do about z? this is a method for 2-space? how does one write a circle in 3-space? :oops:
 
so i guess the equation is a sphere centered at (1/2,1/2,1/2) with radius sqrt(7/4).

i guess i'm having a hard time visualizing how a wire can look like a sphere.

what does the plane x + y + z = 0 look like?

would it be appropriate to now just consider two directions and parametrize with respect to t? ie, x = rcost, y = rsint, z = 0?
 
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