your definition of the integral is off. it isWhen it comes to integration, you have:
lim_Δx->0 Σ_n f(x_n)Δx = ∫f(x)dx
yes, something like that. however, depends on n. so we think of it as a function of n, and consider when ...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.
not necessarily.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?
well, to really get into the meaning of the integral, you have to do analysis. what math are you doing now?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.
mmmm, not really... that is a good conceptualization of what the integral does, but in terms of physical applications, it is perhaps better to interpret the integral in terms of the derivative. that is, it is the reverse of the derivative (which happens to have many applications).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?
this is true, see abovefor 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.
the proof of this in general is really a proof for the fundamental theorem of calculus. it should be in your text as well as countless sites on the net. some sites make a distinction between "the fundamental theorem of calculus" and "the second fundamental theorem of calculus." so look that up as well. based on your questions before, i believe you'd be more interested in "the second fundamental theorem..."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?
for an arbitrary function, it is easy to see this is true.
say f(x) = x
differentiate it we get 1
integrate it, we get back x
so it works.
however, there is what we call the indefinate integral and the definate integral. in the indefinate, we add a constant C to cover the possibility there was a part of the function lost in differentiating, such as constants.
for instance, if the function was f(x) = x + 1, the derivative would again be 1 and the integral would again be x, but you see we lost a 1. so we write the integral as x + C, to account for the possibility of a lost constant. so look out for that as well