Indefinite integral and Fundalmental of calculus?

• pivoxa15
In summary, the conversation discusses the connection between the indefinite integral and the fundamental theorem of calculus (1st part). The speaker believes they are essentially the same, but the second speaker points out that there is a difference between an "integral" and a "theorem". The first speaker then clarifies that the indefinite integral is mainly a notational convenience, while the first part of the FT is the actual calculation of integrals over an interval. The second part of the FT states that the definite integral can be found by evaluating the indefinite integral at the limits of integration and subtracting. The conversation also mentions that there are two parts to the FT, with the first speaker only referring to the second part.
pivoxa15

Homework Statement

What is the connection between the Indefinite integral and the Fundalmental theorem of calculus (1st part)?

The Attempt at a Solution

They are the same to me but the FT is more formal.

You don't think there is any difference between an "integral" and a "theorem"? It makes no sense at all to say "they are the same". That's like saying a solution to a quadratic equation and the quadratic formula are "the same"! You have all the information you need. Now you need to think!

Had a think. It seems to me that the indefinite integral is merely a notational convinience and not linked to anything else. The first part of the FT is the real deal (because integrals must be evaluated over an interval) and to simply the ideas, we introduce the indefinite integral to compute antiderivatives.

The fundamental theorem of calculus essentially says that you can find the definite integral (defined in terms of Riemann sums) by evaluating the indefinite integral at the limits of integration and subtracting.

There are two parts to the FT. You were only referring to the second part?

What is an indefinite integral?

An indefinite integral is an expression in calculus that represents the antiderivative of a function. It is also known as a primitive function. It is used to find the original function when only its derivative is known.

What is the fundamental theorem of calculus?

The fundamental theorem of calculus is a theorem that connects the concepts of differentiation and integration. It states that if a function f(x) is continuous on an interval [a, b] and F(x) is an antiderivative of f(x) on [a, b], then the definite integral of f(x) on [a, b] is equal to F(b) - F(a). In other words, the definite integral of a function is equal to the difference between its antiderivative evaluated at the upper and lower bounds of the interval.

How do you solve an indefinite integral?

To solve an indefinite integral, you need to use integration techniques such as substitution, integration by parts, and partial fractions. First, identify the function and its limits. Then, use the appropriate integration technique to find the antiderivative of the function. Finally, add the constant of integration to get the final answer.

What is the difference between indefinite and definite integrals?

The main difference between indefinite and definite integrals is that indefinite integrals have no specified limits, while definite integrals have upper and lower bounds. Indefinite integrals represent a family of functions, while definite integrals give a specific numerical value. Additionally, definite integrals can be used to calculate area under a curve or find the net change in a function, while indefinite integrals are used to find the original function.

Why is the fundamental theorem of calculus important?

The fundamental theorem of calculus is important because it provides a connection between the two fundamental concepts of calculus - differentiation and integration. This theorem allows us to calculate definite integrals without using Riemann sums, making it an efficient and powerful tool in solving mathematical problems. It also helps us to better understand the relationship between a function and its antiderivative.

• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
11
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
11
Views
988
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
15
Views
914
• Calculus and Beyond Homework Help
Replies
10
Views
1K
• Calculus and Beyond Homework Help
Replies
12
Views
3K