Limit of the Integral Sum

In summary, the author suggests that if you are trying to integrate a function and the limit is not clear, you can partition the interval and take the sum of the series for each partition.
  • #1
siddharth
Homework Helper
Gold Member
1,143
0
To integrate functions as the limit of an integral sum, how can we know which way to take the partition points in the interval?

For example, in
[tex] \int_{a}^{b} x dx [/tex]

I can take the partition points as

[tex] x_o = a [/tex]
[tex] x_1 = a + \delta x [/tex]
...
[tex] x_k = a+ k\delta x[/tex]
where [tex] \delta x = \frac{b-a}{n} [/tex]

So that the sum is [tex] \sum_{k=1}^{n} f(x_{k-1}) \delta x [/tex]

But to integrate
[tex] \int_{a}^{b} \sqrt{x} dx [/tex]

If I take the partition points as above the sum will be
[tex] (\delta x)(\sqrt{a} + \sqrt{a+ \delta x} + ... ) [/tex]

which I cannot find.

I can solve the question if I take the partition points as
[tex] x_0 = a [/tex]
[tex] x_1 = aq [/tex]
...
[tex] x_k=aq^k [/tex]

Where [tex] q=(\frac{b}{a})^(1/n) [/tex].
{The idea to take it this way was given as a hint in the book}

So, is there any other specific manner in which I should spilt the partition points and if so is there a general method in which I can know how to take the values of x_0, x_1, x_2 to solve the problem?
 
Physics news on Phys.org
  • #2
the choice of partition depends upon the function. experience is the thing you need and an educated guess. Ideally you want to choose the partition such that the resulting series is one you can sum. egn if tou wre doing sqrt(x) from 0 to 1 then picking the partition i^2/n^2 as i goes from 0 to n is good cos the square root spits out i/n.
 
Last edited:
  • #3
Are you talking about a numerical integration? In that case, there are two things you can do:(1) take the points equally spaced so the calculations are easier or (2) take the points closer together where the function changes rapidly (i.e. where the derivative is large) so that the integration is more accurate.

Of course, if you are talking about the Riemann sums used to define the integral, then it doesn't matter. As long as the function is integrable, any partition will give the same answer.
 
  • #4
Ah, but proving that something is riemann integrable by the criterion of integrability requiers you to chose the partition family intellignetly, in my experience of teaching analysis. actually to be honest, for any example in the course or the exam there was always a strongly suggested hint.

the approximate rules is pick a partiiton so that the length of the interval and the max.min value spat out are such that when you multiply them together you are addding up r^k from 1 to n times some constant and then appeal to seom result about summing these series.

of course the sensilbe person simply cites that any continuous or montone function is riemann integrable.
 

1. What is the limit of an integral sum?

The limit of an integral sum is a mathematical concept that represents the value that the sum of a function approaches as the number of intervals used in the approximation approaches infinity.

2. How is the limit of an integral sum calculated?

The limit of an integral sum is calculated by taking the limit of the Riemann sum, which is a summation of the areas of rectangles under a curve. This limit is calculated using calculus and can be represented by an integral.

3. What does the limit of an integral sum tell us about a function?

The limit of an integral sum tells us about the overall behavior of a function. It can help us determine if a function is converging or diverging, and can also provide insight into the rate of change of the function.

4. Can the limit of an integral sum be used to find the exact value of a function?

No, the limit of an integral sum is an approximation of the exact value of a function. As the number of intervals used in the approximation increases, the approximation becomes more accurate, but it will never be an exact value.

5. What is the importance of the limit of an integral sum in calculus?

The limit of an integral sum is a fundamental concept in calculus that is used to find the area under a curve. It is also used to solve various problems involving rates of change, optimization, and other real-world applications.

Similar threads

Replies
2
Views
1K
Replies
16
Views
2K
Replies
7
Views
1K
Replies
3
Views
2K
Replies
1
Views
831
Replies
24
Views
2K
Replies
3
Views
929
Replies
12
Views
1K
Replies
10
Views
628
Back
Top