Surface Integral

[cosec(x)]

Hey, I want to check the answer to a question, its not in the book. Its a Surface Integral


Over the region S, the part of the sphere that lies above the cone

Using spherical substitution (phi is angle from z axis naturally ) I get



=0

that doesn't seem right but is it?

[enwiabe]

You want to have it in terms of dpd(theta), not d(phi)d(theta). Need to take into account depth.

[cosec(x)]

you still get a term
anyway, does that coordinate substitution work as we are working with a surface of constant radius. Thus theta and phi give the surface and the depth

[enwiabe]

Nope you're not working with constant radius because the bit in between the cone and the upper hemisphere is being traced between p-values of sqrt(2)/2 and 1.

[enwiabe]

Hmm I was considering the hemisphere as a whole surface btw. Because if you're tracing the OUTSIDE of that surface, then at the z-intercept p is sqrt(2)/2

[cosec(x)]

Yes, the radius is constant, you are integrating the surface of a sphere (a sphere has a fixed radius) that is bounded by a cone, ie. the sphere is cut along the circle where phi=pi/4, and we find that surface, given that it is altered by funtion f=xyz. The surface is a constant distance from the origin, therefore r=1

[enwiabe]

Ohhh is this a closed potato or open potato? Because if you're counting the base of the object...

[cosec(x)]

No, its just a part of the sphere, above, not of the cone, so its the surface of an open "weirdo potato shape"