1. Integration: couple of questions

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?

2. Originally Posted by Greengoblin
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.
nice explanation. your thoughts are more or less accurate.

When it comes to integration, you have:

lim_Δx->0 Σ_n f(x_n)Δx = ∫f(x)dx
your definition of the integral is off. it is $\lim_{n \to \infty} \sum_{i = 1}^{n} f(x_i) \Delta x = \int f(x)~dx$

..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.
yes, something like that. however, $\Delta x$ depends on n. so we think of it as a function of n, and consider when $n \to \infty$.

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?
not necessarily.

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.
well, to really get into the meaning of the integral, you have to do analysis. what math are you doing now?

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?
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).

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.
this is true, see above

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?
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..."

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

3. Thanks. Are you saying that for an integral, Δx=f(n)? I don't understand why Δx would be a function of n, and why you can consider n approaching infinity to mean Δx approaching 0 - and Δx must approach 0 for the width of each column to become infinitesimally small?

well, to really get into the meaning of the integral, you have to do analysis. what math are you doing now?
I've just finished learning about differentiation, and now I'm trying to learn about integration too. By "Analysis" are you talking about things like sequences and series or more advanced? I've just started looking at sequences and series too, with arithmetic progression and stuff. Actually I'm not sure where the formula for an arithmetic series comes from. I understand that nth term = a + (n-1)d, but I don't understand how this can lead to the formula for the series:

Sn = n/2[2a + (n-1)d]

for an arbitrary function, it is easy to see this is true.

say f(x) = x
But f(x)=x isn't a general function is it? I mean like a proof for any function f(x), that could prove it true no matter what the function is. For example it could aslo be trig, or exponential or polynomial etc. Thanks for the reply

4. Originally Posted by Greengoblin
Thanks. Are you saying that for an integral, Δx=f(n)? I don't understand why Δx would be a function of n, and why you can consider n approaching infinity to mean Δx approaching 0 - and Δx must approach 0 for the width of each column to become infinitesimally small?
yes, $\Delta x \to 0$ as $n \to \infty$, but it is just apart of the definition of the integral to use only the limit with respect to n (imagine having to use a double limit in the definition! one as $\Delta x \to 0$ and one as $n \to \infty$). it is easier to do one. and it is easy to write $\Delta x$ in terms of n and the end points of the interval we are considering, so why not do it that way?

I've just finished learning about differentiation, and now I'm trying to learn about integration too. By "Analysis" are you talking about things like sequences and series or more advanced?
more advanced. analysis is something you do after your elementary calculus sequence (calculus 1,2, and 3).

I've just started looking at sequences and series too, with arithmetic progression and stuff. Actually I'm not sure where the formula for an arithmetic series comes from. I understand that nth term = a + (n-1)d, but I don't understand how this can lead to the formula for the series:

Sn = n/2[2a + (n-1)d]
you can see here for how the sum is derived.

you can also try do derive the formula using the intuitive approach it talks about.

say you want to sum the first n terms, that is, find:

$S_n = 1 + 2 + 3 + \cdots + (n - 2) + (n - 1) + n$

notice that if we add the first and last terms, and then the second and second to last terms, and third and third to last terms, we get n + 1 in each case, as illustrated below:

$1 + n = n + 1$

$2 + (n - 1) = n + 1$

$3 + (n - 2) = n + 1$

.
.
.

and so on

so the sum of the n terms amounts to summing all these (n + 1)'s. how many are there? well, they result from us pairing off terms, so there must be half the number of (n + 1)'s as there are the number of terms. there are n terms, so there are $\frac n2$ (n + 1)'s. thus:

$S_n = \frac n2(n + 1)$

here 1 is the first term, analogous therefore to $a_1$ and n is the nth term, analogous therefore to $a_n$. so we can replace them.

so, $S_n = \frac n2(a_1 + a_n)$

replacing $a_n$ with the formula for an arithmetic series yields the formula you mentioned earlier.

by no means is this a rigorous proof. here i showed this for a particular kind of arithmetic series (one in which the first term was 1 and the common difference is 1). however, this procedure can be generalized to any term. i was merely trying to give you an intuitive feel for how the formula came about. the link i gave you shows another way to derive the formula as well.

But f(x)=x isn't a general function is it? I mean like a proof for any function f(x), that could prove it true no matter what the function is. For example it could aslo be trig, or exponential or polynomial etc. Thanks for the reply
as i said, what you are asking for, in general, is the proof to the fundamental theorem of calculus. you can look it up on wikipedia or in your text. every calculus text has it. if yours doesn't, it should be burnt.