Hi, I have a couple of questions on integration:
1.) Expressing the derivative of a function f(x) from the limit gives you:
lim_Δx->0 [f(x+Δx) - f(x)] / Δx = d/dx f(x)
..so that you essentially have a small change in f(x) divided by a small change in x, to give you the gradient of a secant line, which approches the actual value of the gradient at a point, when Δx approaches 0. I find this easy to follow, and like to think of the gradient in terms of a 'rate of change' of one variable, y, with another, x, when thinking about physical applications...e.g. the rate of change of distance with time, or velocity with pressure.
When it comes to integration, you have:
lim_Δx->0 Σ_n f(x_n)Δx = ∫f(x)dx
..so that for each 'n', you have a value f(x_n)Δx for the area of a narrow column, and by adding these up you get a total value for the area under the curve, which gets closer and closer to the real value of this area, as Δx approaches 0.
What I don't get is the multiplication by the term dx. if the value of dx is infinitessimally small, then musn't the product of dx, and the integrand be infinitesimally small also? For the derivative, dy/dx, It still doesn't make much sense to divide two infinitesimally small values, but I can at least understand this as the ratio of Δy and Δx (gradient) becoming more accurate, as the lengths approach 0. For the integral, I'm struggling for a nicer interpretaion of what it means.
It's all good knowing that integration gives you a function describing "the area under a graph", but in a physical sense is there a better interpretation?..for example, I can think of derivatives as a rate of change of one quantity with another, which is very easily translated to physical applications, but integrals seem harder to place.
2.) If integration and differentiation are opposite processes, can someone show me the proof of this for an arbitrary function? Do you have to do this from the limit or something?