Expert: Scotto - 8/4/2010

Question
QUESTION: sir,
what is the proof that riemann summation(limit n --> infinity, sigma i=1 to n (lenth of each interval*value of function at that interval)) equals to definite integral of that function...
sorry for not using symbols due to unavailability. i think you will understand it by name riemann summation.

      thank you

ANSWER: This is because the Riemann Sum aprroximatets the area of each retctangle as
(C-A)(f(A)+4f(B)+f(C))/6 where A is the left x-value, B is the center x-value,
and C is the right x-value.  This is 6 x-values that all give the height of the
rectangle.  The best way to approximate the height is to divide the sum of these
6 values by 6 and multiply by the width, which is C-A.

It can be seen that the smaller C-A gets, the closer all of the f values on that interval get to
each other, so the closer the area is to the base times the sum of six heights over 6 since that is the same as saying that the area of each rectangle is approximately the base times the height.


---------- FOLLOW-UP ----------

QUESTION: sorry sir,
i know how the riemanns summation gives the area bounded by curve and one of the axes. but i am asking that how this area is obtained by integrating the function. i know that by integrating we are finding the anti derivative of that function and by substituting the limits we get the area.
but how?

thankyou sir

Answer
It all depends on the function that is being looked at.

For derivatives, all continuous functions come down to some multiple, product, and/or quotient of simple functions.
There is a product rule and a quotient rule for these
if they are more involved.

For integrals, however, it gets rather complicated.  If we have
f(x), usually F(x) is used to denote the integral.  There are
entire books written on the evaluation of these integralsm, for
depending on the function involved, the integral is different.  
Taking derivatives can be summarized in derivatives of
polynomials, e^x, and trig functions.

Thus, there are the following basic integrals:
f(x) = ax, then F(x) = ax²/2 + C, where C is a unknown constant
f(x) = ax^n, then F(x) = [ax^(n+1)]/(n+1) + C
f(x) = e^x, then F(x) = e^x + C
f(x) = sin(x), then F(x) = -cos(x) + C
f(x) = cos(x), then F(x) = sin(x) + C

This is just a few (namely, 5) of the basic integrals.
There is no simple way to do them but to just know them.

Note that if f(x) = e^x, the derivative and integral are both e^x.
If the inverse is looked at, that's g(x) = ln(x), it is not quite
so simple.  It is known that g'(x) = 1/x abd G(x) = x*ln(x) - x + C.

For sin(x) and cos(x), just think about the alphabetical order.
Note cos(x) comes before sin(x) since 'c' comes before 's', so the integral is negative.  It can also be thought of as the reverse of
the derivative.

The place where integral arise with no C is when there are limits involved.  The derivative finds the slope at a point and the integral finds the area beneath the curve between x=a and x=b.
A problem with the endpoints declared looks like
|b
|a f(x) dx = F(b) - F(a) where | is used for the integral sign.

I know, it is suppose to be curvy, but I can't find anything like that in the character set.  It looks a whole lot better in the picture I am sending.