Question

Question about function-level multiplication (Integration):

Answer 1

Let's say we're given a spring attached to an object and a wall, pushing the object further further beyond the spring's natural stretch gradually increases the required force to continue pushing the object by the given pattern:
F(x) = kx, where F(x) is the Force required, k is the spring stretching ratio, and x is the distance.
If we want to find the Work... through normal means
W = F * d, but because this is a function, we need to ripple carry the effect of multiplying the "first" element to the next and so on, so we use integration.
so we zoom in, \[\lim_{dx \rightarrow \infty}(\sum_{i=0}^{n}(F(x_i) *dx) = \int\limits_{a}^{b}(F(x)dx\]
My question is: this is dependent on the very small interval considered.
What if I had to function-multiply something like...
\[C=2\pi r -> C(t) = 2\pi r(t)\] where the radius is now a fucntion of time?
How do I perform the conversion to integration when dt is not even considered?

Answer 2

that summation integration formula is not quite right

Answer 3

oh no, I meant that dx approaches 0!!