Question about definite and indefinite integrals

I may be confusing you by giving you more information than you asked for". But, to answer your question, no, you have not told me anything confusing. Everything you have said has been relevant and helpful in understanding the concept of indefinite integrals.
  • #1
jonjacson
447
38
First, just to check, I write what I think and let me know if I am wrong:

The definite integral of a function gives us a number whose geometric meaning is the area under the curve between two limiting points.
We can calculate this integral as the limit of the sum of the rectangles and the integral symbol means a sum over all rectangles.

The indefinite integral of the function f(x) is the function primitive F(x) whose derivative is f(x), and here we are using the fundamental theorem of calculus. Why do we use the same symbol?

My question is, How could we calculate the indefinite integral without the fundamental theorem?

Could we try it using the definite integral between the limits 0 and x?

From a geometric point of view, How could we talk about the indefinite integral?

Thanks!
 
Last edited:
Physics news on Phys.org
  • #2
jonjacson said:
Why do we use the same symbol?
Because basically it's the same thing. Compare using the same term ##\sin ## for the sine function and for the value of the sine function at some point ##x##.
jonjacson said:
How could we calculate the indefinite integral without the fundamental theorem?
We could create a table of F values from the sum of rectangle areas, but we would still need to state where we start integrating (i.e. the table would list ##F(x)-F(x_{\rm start}## ). So: Yes to your
jonjacson said:
Could we try it using the definite integral between the limits 0 and x?
Note: this doesn't give you a function in the from of a 'recipe', though.
jonjacson said:
From a geometric point of view, How could we talk about the indefinite integral?
It tells you how the area under the curve varies with the independent variable
 
  • #3
jonjacson said:
First, just to check, I write what I think and let me know if I am wrong:

The definite integral of a function gives us a number whose geometric meaning is the area under the curve between two limiting points.
We can calculate this integral as the limit of the sum of the rectangles and the integral symbol means a sum over all rectangles.

The indefinite integral of the function f(x) is the function primitive F(x) whose derivative is f(x), and here we are using the fundamental theorem of calculus. Why do we use the same symbol?

My question is, How could we calculate the indefinite integral without the fundamental theorem?

Could we try it using the definite integral between the limits 0 and x?

From a geometric point of view, How could we talk about the indefinite integral?

Thanks!

You seem to be largely answering your own questions here. The only thing I would add is that you could see ##F## as a function which gives you definite integral between two points:

##\int_{a}^{b}f(x)dx = F(b) - F(a)##

And, it turns out that any ##F## where ##F' = f## does the job. And, you call any such ##F## an indefinite integral of ##f##. More precisely, the set of functions ##\lbrace F: F' = f \rbrace## is the indefinite integral.
 
  • Like
Likes BvU
  • #4
BvU said:
Because basically it's the same thing. Compare using the same term ##\sin ## for the sine function and for the value of the sine function at some point ##x##.
We could create a table of F values from the sum of rectangle areas, but we would still need to state where we start integrating (i.e. the table would list ##F(x)-F(x_{\rm start}## ). So: Yes to your
Note: this doesn't give you a function in the from of a 'recipe', though.
It tells you how the area under the curve varies with the independent variable

The indefinite integral gives you the area (if you substitute for particular values), not the rate of change of the area, which is the function itself right?
PeroK said:
You seem to be largely answering your own questions here. The only thing I would add is that you could see ##F## as a function which gives you definite integral between two points:

##\int_{a}^{b}f(x)dx = F(b) - F(a)##

And, it turns out that any ##F## where ##F' = f## does the job. And, you call any such ##F## an indefinite integral of ##f##. More precisely, the set of functions ##\lbrace F: F' = f \rbrace## is the indefinite integral.

THanks.
 
  • #5
jonjacson said:
The indefinite integral gives you the area (if you substitute for particular values), not the rate of change of the area, which is the function itself right?
Yes.
Example: constant function. Indefinite integral: x itself, linear function. Area changes linearly with the independent variable.
 
  • #6
BvU said:
Yes.
Example: constant function. Indefinite integral: x itself, linear function. Area changes linearly with the independent variable.
I think there was a misinterpretation.

When you talked about changed linearly, you talked about the diferent values a function primitive F(x) can take, I thought you meant its rate of change F'(x) , that is why I said the rate of change was the funcion f(x) itself.

I guess!
 
  • #8
Just to add to the confusion: In many cases it is possible to evaluate the definite integral but not the indefinite...
 
  • #9
pwsnafu said:
You might be interested in the Risch Algorithm.
Very interesting yes.
Svein said:
Just to add to the confusion: In many cases it is possible to evaluate the definite integral but not the indefinite...

I understand that. Finding an area is different to finding a function whose differential is a given one.

Have I told you anything confusing?
 
  • #10
jonjacson said:
Have I told you anything confusing?
Just an expression. Means: "This information that was not asked for and may not be very relevant"
 

Related to Question about definite and indefinite integrals

1. What is the difference between definite and indefinite integrals?

Definite and indefinite integrals are both mathematical operations that involve finding the area under a curve. However, the main difference is that definite integrals have specific limits of integration (numbers that define the start and end points of the curve), while indefinite integrals do not.

2. How do you solve a definite integral?

To solve a definite integral, you first need to determine the limits of integration. Then, you can use various techniques such as the fundamental theorem of calculus, substitution, or integration by parts to evaluate the integral. Finally, plug in the limits of integration to get the numerical value of the definite integral.

3. What is the purpose of using definite integrals?

Definite integrals have a variety of applications in science and engineering, such as calculating the area under a curve, finding volumes and areas of irregular shapes, and determining the average value of a function. They are also used to solve problems in physics, economics, and other fields.

4. Can you give an example of a definite integral?

Sure, an example of a definite integral is ∫02 x² dx. This integral represents the area under the curve y = x² from x = 0 to x = 2. Evaluating this integral gives us the answer of 2.67.

5. What is the difference between indefinite integrals and antiderivatives?

Indefinite integrals and antiderivatives are often used interchangeably, but they are not exactly the same. An indefinite integral is the set of all possible antiderivatives of a function, while an antiderivative is a specific function that, when differentiated, gives the original function. In other words, an antiderivative is a specific solution to an indefinite integral.

Similar threads

Replies
20
Views
2K
Replies
10
Views
2K
Replies
31
Views
1K
Replies
10
Views
4K
Replies
8
Views
579
  • Calculus
Replies
6
Views
2K
Replies
6
Views
1K
Replies
4
Views
3K
Replies
4
Views
2K
Back
Top