what is the difference between summition and integration?

Answer 1

An infinite sum is the limit of a finite sum as the number of summands goes to infinity: e.g. 1+0.1+0.01+0.001+0001+00001+.... = (sum of 0.1^k as k goes from 0 to infinity) = 10/9.

A Riemann-integral of a function f on the interval [a,b] is also a limit of a SPECIAL sum:

You divide the interval [a,b] into subintervals by creating points a = x0 < x1 < x2 <...< xn = b. Then choose a point y_k in every subinterval [x_k, x_(k+1)) (k = 0,1,...,n-1), and take the Riemann-sum S = sum as k goes from 0 to n-1 of f(y_k)*(x_(k+1) - x_k).

This Riemann-sum is the sum of the selected values y_k in each subinterval times the length of the subinterval, and it can be interpreted as the total area of approximating rectangles of the function, where the rectangles have height y_k and base (x_(k+1)-x_k).

Now, if the number of subintervals goes to infinity such that the length of each subinterval goes to 0, and the Riemann-sum has a limit which is the same regardless of the choice of the subintervals and the interior points, then this limit is called the Riemann-integral of f in the interval [a,b].

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Another more advanced approach is Lebesgue-integration. In this setting, instead of dividing the interval [a,b] on the x-axis into subintervals, you create subintervals on the y-axis. The Lebesgue-integral is perhaps less intuitive but it is much easier theoretically to prove some theorems in this setting (e.g. for exchanging limits and integrals) and more functions become integrable than in the Riemann setting.

The Lebesgue-integral is first defined for non-negative step functions (which are piecewise constant and have a finite number of jumps), where it is just a simple sum of a finite number of terms, then it is extended to non-negative measurable functions (as the least upper bound of the integrals of all step-functions below the function), and finally to arbitrary measurable functions (as the integral of their positive part minus the integral of the negative part).

Interestingly, an infinite sum is a special Lebesgue-integral (whereas a Riemann-integral was a special infinite sum).

Answer 2

Integration is the limit of the summation process to find the area under a curve. In using the trapezoidal method, you break the x-axis into many small pieces of width w, calculate the value of some function (say f(x)) at nw and (nw+w), then sum up each of the areas you find in this way to get the total area. Integration is what you get in the limit as w -> 0. In fact, the integral sign itself reflects this since it is an elongated S (first used by K.F. Gauß)


Doug

Answer 3

Summataion is done over entire intervals where each subinterval is of finite length, usually of integral lengths. Also natural counting numbers are used. For example, if I am summing up 1/n as n goes from 1 to 10, this is like drawing 9 rectangles that all have width one and height 1/n (depending if you taking the upper sum or the lower sum). Adding up all of the 9 rectangles' area will give me a very rough estimate of the actual area underneath the curve 1/n from 1 to 10. I can also get a bound on the actual area by finding out the upper sum and then the lower sum. I know that the actual area is somewhere between the two.

The smaller I make the width of each rectanlge, the better my estimate will get for both the upper sum and the lower sum. An as the width of my rectanlges gets smaller and smaller, the number of actual rectangles between 1 and 10 will increase.

In integration, what you have done is that the width of your rectangle is infinitesimally small (dn in this case) and the number of rectangles is infinite, which gives you the actual area. Integration can be done from anywhere to anywhere. It is not limited to natural numbers for example. It is not limited to whole numbers. Because remember, in summation the index is actually a counting number starting from zero or one and then you increase it by one. In integration, you are considering everything. The integral also gives you the area that is below the x-axis as a negative number.

So if I integrate xdx from -1 to 1, the integral will be zero because the are from -1 to 0 cancels out the area from 0 to 1. So if you want to find out the actual area, then make sure to find out when the curve is actually above or below the x-axis, split up the integral into as many parts as you want and then take their absolute values and add them up.

I don't know if you have done infinite series or the convergence tests or not but if you remember, the integral test is one of them. The integral and summation are close enough to tell if they both converge or diverge but their values are too different to be estimated from each other.

Answer 4

Essentially the number and size of the added terms. Summation will add discrete values of a varying magnitute; however when the number of discrete values reach infinity and the size of the terms goes to zero (dx), you have an integration.

Answer 5

summation of f(x), for x = 0 to x = n
s = ∑x (i=0 to n)

Integration of f(x) is,
∫f(x) = lim {∆x →0} ∑f(x)∆x