 #1
snatchingthepi
 148
 38
 Homework Statement:
 Finding the volume above the cone z = sqrt(x^2 = y^2) and below sphere x^2 + y^2 + z^2 = 1
 Relevant Equations:

Cone: z = sqrt(x^2 = y^2)
Sphere: x^2 + y^2 + z^2 = 1
So I can push this integral all the way to the end and see I get a negative volume.
I solve for the intercepts of the cone and sphere at r^2 = 1/2. Seeing this cone is inside the sphere and the sphere is around it, I figure I should integrate from sqrt(1/2) to 1 since we're dealing with a unit sphere, and these limits would give the area *above* the cone and *below* the sphere.
But when I check the solution manual I see the problem involves using radial limits from 0 to sqrt(1/2) but the equation that is integrated is exactly the same as mine. Using limits from 0 to root(1/2) would seem to me to give the area above the plane projection of the sphere and below the cone, not above the cone and below the sphere. Can someone help me understand where this is coming from?
I solve for the intercepts of the cone and sphere at r^2 = 1/2. Seeing this cone is inside the sphere and the sphere is around it, I figure I should integrate from sqrt(1/2) to 1 since we're dealing with a unit sphere, and these limits would give the area *above* the cone and *below* the sphere.
But when I check the solution manual I see the problem involves using radial limits from 0 to sqrt(1/2) but the equation that is integrated is exactly the same as mine. Using limits from 0 to root(1/2) would seem to me to give the area above the plane projection of the sphere and below the cone, not above the cone and below the sphere. Can someone help me understand where this is coming from?