Re: help with basic integration concept

prof <rbmadden@xxxxxxxx> wrote:
Using the concept that differentiation is the limit of f(x+ dx) -
f(x)/dx it is easy to show, for example, that the derivative of y =
x**3 is 3x**2. However, using the concept of integration as being the
limit of the sum of y(dx), how can one show that the integral of 3x**2
is x**3 without resorting to the fact that integration is simply the
reverse of differentiation.

Start with the fact that the sum of the first n square numbers, that
is the sum of i^2 from i=1 to n, is equal to n(n+1)(2n+1)/6. (If you
didn't know that identity, you can check it quickly by observing
that it's zero when n=0 and that if you work it out for two adjacent
values of n and take the difference you are left with the
appropriate square number. Deriving it from scratch is another
matter, which I can explain too if you need it.)

Suppose we are trying to work out the integral of 3x^2 from x=0 to
A. We can approximate this integral by dividing the range into n
parts so that we get a number of long thin vertical strips; then we
sum the heights of those strips (as approximated by the height of
the curve y=3x^2 at one end of each one), and multiply by A/n to
scale back to the real width of the area we're integrating.

Thus, for a given n, our approximation to the integral is equal to

n
A/n * sum 3(Ai/n)^2
i=1

n
= 3(A/n)^3 * sum i^2
i=1

= 3(A/n)^3 * n(n+1)(2n+1)/6

= 3(A/n)^3 * (n^3/3 + n^2/2 + n/6)

= A^3 + 3A^3/2n + A^3/2n^2

and it's clear that as n tends to infinity, the last two terms tend
to zero and we are left with the integral of 3x^2 from x=0 to A
being exactly A^3.

(This is not a properly rigorous demonstration. To be completely
solid I should have given two approximate sums, one of which was
known to be larger than the real integral and one of which was known
to be smaller, and shown that they both converged to the same limit,
which would have actually proved that the integral _exists_ as well
as showing what its value would be if it did. The additional work is
left as an exercise for the reader.)
--
Simon Tatham "The distinction between the enlightened and the
<anakin@xxxxxxxxx> terminally confused is only apparent to the latter."
.

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