I'm trying to compute the area of the Cone (Z = sqrt[ x^2 + y^2] below the plane z = 2.
Using cylindricl polars coordinates:
phi: [ -pi/2, pi/2]
r: [ 0, 2]
z: [ r, 2]
with the Jacobian being r.
I get 4/3 pi which I think is the correct answer.
However, using Spherical polar coordinates and using the following limits:
Phi: [ -pi/2, pi/2 ]
Theta: [ 0, pi/4 ]
r: [ 0, 2 ]
with the Jacobian being r^2 sin (theta) I'm getting a different answer. Could anyone tell me if my limits are correct?
I think I know where the limits comes from:
Unlike for a sphere the r is changes, as we track round with theta changing form zero to pi/4 the distance form the centre isn't constant.
Here r it is given by the hypotenuse of a triangle.
Cos (theta) = adjacent/hyp = z/r Where z =2, so we have;
r = 2/cos(theta)