Multi variable calculus

area of a surface? idk yet future me whats the answer

kind of like a line integral without the function… what is bprp up to?

As a Calc 1 student this doesnt make any sense

It depends on what the limits a and b are representing.
Two ways of considering it:
1.
Let x, y and z the coordinates of a point M in space with respect to an ortho- normal coordinating system with origin O, so is
L=||OM|| =sqrt(x^2 + y^2 + z^2), the norm/magnitue of vector OM, i.e., the length of vector OM
=> dL=d(|OM||) =sqrt(dx^2 + dy^2 + dz^2)
=>∫[√(…)]dt = ∫[dL/|dt|]dt
=> ∫[√(…)]dt = sign(dt) ∫dL

Assuming the point M describes a travel trajectory T of an object over time t, and a and b represent the position on the trajectory f at two different times t1 and t2, t1<t2, then
dt>0, and
∫[√(…)]dt = ∫dL = travel distance along trajectory T from a to b

In the simplest case of linear trajectory
=> ∫[√(…)]dt = (b-a)

2. Alternatively:
Assuming the point M=(x(t), y(t), z(t)) describes a trajectory T of an object traveling with a speed V(t)=[vx(t), vy(t), vz(t)] and a and b represent the positions on the trajectory T at two different times t1 and t2, t1<t2, then
vx(t)=dx/dt
vy(t)=dy/dt
vz(t)=dz/dt
=>
√[(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2] = ||V(t)||
=> [√(…)]dt = ||V(t)||dt is the travel distance dL during the travel time dt
=> ∫[√(…)]dt = ∫dLdt, is the travel distance from a to b along the trajectory T

Arc length formula for three dimensions?

Vector calculus expanded. Root vector v squared integrated from time a to time b. His is the forward approach, mine is the backward!