What are DS and s in curve integral? What do they mean? What's the big change, the small change, the constant change, the approximate sum and the limit?

S is the integral variable, DS is equivalent to the increment of the variable. Because the physical meaning of curve integral represents the quality of the curve. Previously, we knew that the quality formula of a curve is the length of the curve multiplied by the density of its unit length. However, this is applicable to curves with uniform quality distribution. In reality, most of the curves we encounter are not uniform, so we have a problem
To solve this problem, the method is curve integral
We can transform this into the curve of uniform mass distribution that we have learned, which will be used. Divide a long non-uniform curve into very small segments (assuming that the length of each segment is DX), so that each small segment can be approximately regarded as uniform, and then we can use the above formula, That is to say, "large to small"; to replace a non-uniform section with a uniform one, and to replace the variable with a constant is "constant to variable"; and then to add up the mass of each small section is the quality we want; but do we find that there is still a little error in the above solution? After all, we can replace the variable with a constant, but we can do approximate substitution as long as we divide the curve small enough, The actual mass is very close to the mass we solved, which is "approximate sum". Of course, the finer the solution, the better. This requires that each segment be small enough to the limit, that is, the length ds of each segment is close to 0, which is "limit"

How to find DS? Method for the first kind of curve integral?

cosαds=dx
cosβds=dy
cosγds=dz
α. β and γ are the angles between the curve and x-axis, Y-axis and z-axis respectively
Generally, we don't need to find DS, but we can solve the problem by using the formula
I=∫[L]f(x,y,z)ds=∫[a,b]f(x(t),y(t),z(t))sqrt[(x'(t))^2+(y'(t))^2+(z'(t))^2]dt

Find the surface integral ∫ ∫ ∑ (y + X + Z) ds, where ∑ is the sphere x ^ 2 + y ^ 2 + Z ^ 2 = z > = H (0) on a ^ 2

Soga, where don't you understand? Just draw a picture

Let ∑ be a sphere x ^ 2 + y ^ 2 + Z ^ 2 = 4, then the curved area is divided into ∮ (x ^ 2 + y ^ 2 + Z ^ 2) ds=
I calculate DS = 2 / (4-x ^ 2-y ^ 2) ^ 1 / 2dxdy
∫∫ (x ^ 2 + y ^ 2 + Z ^ 2) ds = x ^ 2 + y ^ 2 + Z ^ 2) ds = ∫∫ 4.2 / (4-x ^ 2-y ^ 2) ^ 1 / 2dxdy, I just don't understand polar coordinate transformation. By the way, there's another way to substitute it into spherical equation ∫ (x ^ 2 + y ^ 2 + Z ^ 2) ds = ∫∫ 4ds = 4 * 4 π R ^ 2 = 16 π R ^ 2 = 64 π, some don't understand it. Thank you for your help

∫ ∫ 4.2 / (4-x ^ 2-y ^ 2) ^ 1 / 2dxdy in this step, we should find the integral region, which is the projection of the sphere on the xoy plane, that is, x ^ 2 + y ^ 2 = 4
Let x = RCOs θ, y = rsin θ in polar coordinates, then - π ≤ θ ≤ π, 0 ≤ R ≤ 2, DXDY = rdrd θ can be substituted into integral
In the sphere x ^ 2 + y ^ 2 + Z ^ 2 = 4, obviously x ^ 2 + y ^ 2 + Z ^ 2 = 4 holds, so ∫ ∫ (x ^ 2 + y ^ 2 + Z ^ 2) ds = ∫ ∫ 4ds. Because the integral is carried out on the sphere, not in the region contained in the surface

Is the last part of the algorithm formula of the first kind of curve integral actually the arc differential formula DS = √ [1 + (dy / DX) ^ 2] * DX?
Why does the arc differential formula DS = √ [1 + (dy / DX) ^ 2] * DX appear? What is the relationship between the first kind of curve integral and it?

For its arc length formula, according to the physical meaning of the first type of curve integral (the mass of the arc length), the integral function represents the density, and DS represents the arc length. When solving the general first type of curve integral, as long as you use the arc length formula to transform the first type of curve integral into the second type of curve integral, the formula you mentioned will appear

What are DS and s in curve integral? What do they mean? What's the big change, the small change, the constant change, the approximate sum and the limit?