What is the projection curve equation of the space curve equations z = x ^ 2 + y ^ 2, z = 2 - (x ^ 2 + y ^ 2) on the xoy plane and how to calculate it, Tell me about the curve equation, such as on that plane, how to spin the segment around which axis, OK,

What is the projection curve equation of the space curve equations z = x ^ 2 + y ^ 2, z = 2 - (x ^ 2 + y ^ 2) on the xoy plane and how to calculate it, Tell me about the curve equation, such as on that plane, how to spin the segment around which axis, OK,

On the xoy plane, the projection is a circular surface, and the space image is two conical side surfaces
When Z is brought in with a specific value, we can get that, for example, when z = 0, the image of X and Y is a point, and when other values are used, the image of X and Y is a circle with increasing radius

The surface integral of high number to area

First of all, the equation of time-sharing integral surface of curved area can be expressed with integral, then integral = ∫ ∫ (x ^ 2 + y ^ 2-x ^ 2-y ^ 2-1) ds=
-∫∫ DS, and the surface integral ∫∫ DS represents the area of the integral surface. In this problem, the cone side area = π / √ 2, so the integral = - π / √ 2

Why is DS = r2sin ψ D ψ D θ not DS = r2d ψ D θ
If the surface is a sphere, it is appropriate to use this expression, but the result is not right. Why

Take a globe and compare it. The higher the latitude (ψ), the shorter the circumference of the latitude, and the smaller the area represented by each longitude (D θ)