Definite Integral & Limit: Proving a Relation

In summary, a definite integral is a mathematical concept used to represent the area under a curve on a graph. It is calculated by finding the anti-derivative of a function and substituting the upper and lower limits. There is a relationship between a definite integral and a limit, where the former can be seen as the limit of a Riemann sum. These two concepts can also be used to prove relations by showing that the limit is equal to the definite integral. In real-life, definite integrals and limits have various applications in fields such as physics, engineering, and economics.
  • #1
dtl42
119
0

Homework Statement


Show that each limit is a definite integral.
[tex]Lim (n \rightarrow \infty) [/tex] of [tex]\sum \frac{n}{n^{2}+i^{2}} [/tex] from [tex]i=1[/tex] to [tex]n[/tex]

Homework Equations


[tex]Lim (n \rightarrow \infty) [/tex] of [tex]\sum f(c)\Delta X [/tex] from [tex]i=1[/tex] to [tex]n[/tex]


The Attempt at a Solution



I can't really get this started, so any help at all would be great.

Thanks.
 
Physics news on Phys.org
  • #2
Try multiplying by [tex]\frac{\frac{1}{n^2}}{\frac{1}{n^2}}[/tex].
 
  • #3
What do you think [itex]\Delta x[/itex] should be?

(My hint is really the same as foxjwill's.)
 

FAQ: Definite Integral & Limit: Proving a Relation

What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve on a graph. It is denoted by the symbol ∫ and has a lower and upper limit that define the range of values to be integrated.

How is a definite integral calculated?

To calculate a definite integral, you first need to find the anti-derivative of the function. Then, you can substitute the upper and lower limits into the anti-derivative and subtract the two values to find the area under the curve.

What is the relationship between a definite integral and a limit?

The relationship between a definite integral and a limit is that the definite integral can be seen as the limit of a Riemann sum, which is a way of approximating the area under a curve by dividing it into smaller rectangles. As the number of rectangles increases to infinity, the Riemann sum approaches the value of the definite integral.

How can a definite integral and a limit be used to prove a relation?

To prove a relation using a definite integral and a limit, you need to show that the limit of the Riemann sum is equal to the definite integral. This can be done by choosing a specific function and limits, and then evaluating both the definite integral and the limit to show that they are equal.

What are some real-life applications of definite integrals and limits?

Definite integrals and limits have many real-life applications, such as calculating the distance traveled by an object with a changing velocity, finding the volume of irregularly shaped objects, and determining the average value of a function over a certain interval. They are also used in economics, physics, and engineering to model and analyze various systems.

Similar threads

Real analysis: prove the limit exists
Replies
13
Views
3K
Evaluate the limit of this harmonic series as n tends to infinity
Replies
17
Views
2K
Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds
Replies
6
Views
1K
Prove that this series converges uniformly on R
Replies
4
Views
996
Expressing a limit as a definite integral
Replies
7
Views
2K
Limit Definition of Derivative as n Approaches Infinity
Replies
20
Views
2K
Limit and Integration of ##f_n (x)##
Replies
8
Views
1K
How to show that these sequences are summable?
Replies
1
Views
1K
Show that the limit (1+z/n)^n=e^z holds
Replies
6
Views
935
Find limit involving square of sine
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top