## Definition

**The fundamental theorem of calculus **is a theorem that links the concept of integrating a function with that of differentiating a function. The fundamental theorem of calculus justifies the procedure by computing the difference between the antiderivative at the upper and lower limits of the integration process. In this article, let us discuss the first, and the second fundamental theorem of calculus, and evaluating the definite integral using the theorems in detail.

### First Fundamental Theorem of Integral Calculus (Part 1)

The first part of the calculus theorem is sometimes called the first fundamental theorem of calculus. It affirms that one of the antiderivatives (may also be called indefinite integral) say F, of some function f, may be obtained as integral of f with a variable bound of integration. From this, we can say that there can be antiderivatives for a continuous function.

**Statement: **Let *f* be a continuous function on the closed interval *[a, b]* and let *A(x)* be the area function. Then *A′(x) = f(x)*, for all *x ∈ [a, b]*.

Or

Let *f* be a continuous real-valued function defined on a closed interval *[a, b]*. Let *F* be the function defined, for all *x* in *[a, b]*, by:

Then *F* is uniformly continuous on *[a, b]* and differentiable on the open interval *(a, b)*, and

*F'(x) = f(x) ∀ x ∈(a, b)*

Here, the *F'(x)* is a derivative function of *F(x)*.

### Second Fundamental Theorem of Integral Calculus (Part 2)

The second fundamental theorem of calculus states that, if the function *f* is continuous on the closed interval *[a, b]*, and *F* is an indefinite integral of a function *f* on *[a, b]*, then the second fundamental theorem of calculus is defined as:

**F(b)- F(a) = ∫ _{a}^{b} f(x) dx**

Here RHS (right-hand side) of the equation indicates the integral of *f(x)* with respect to *x*.

*f(x)* is the integrand.

*dx* is the integrating agent.

*a* indicates the upper limit of the integral and *b* indicates a lower limit of the integral.

The function of a definite integral has a unique value. The definite integral of a function can be described as a limit of a sum. If there is an antiderivative *F* of the function in the interval *[a, b]*, then the definite integral of the function is the difference between the values of *F*, i.e., *F(b) – F(a)*. ^{1}

### Remarks on the Second Fundamental Theorem of Calculus

- The second part of the fundamental theorem of calculus tells us that
*∫*(value of the antiderivative_{a}^{b}f(x) dx =*F*of*f*at the upper limit*b*) – (the same antiderivative value at the lower limit*a*). - This theorem is very beneficial because it provides us with a method of estimating the definite integral more quickly, without determining the sum’s limit.
- In estimating a definite integral, the essential operation is finding a function whose derivative is equal to the integrand. However, this process will reinforce the relationship between differentiation and integration.
- In the expression
*∫*, the function_{a}^{b}f(x) dx*f(x)*or say*f*should be well defined and continuous in the interval*[a, b]*.

## Who

If you are like me, I do not remember having the FTC explained to me very well. Many of these references do a fine explanation to help you truly appreciate the FTC.

The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the integral.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.

The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals.

## What

A definition for derivative, definite integral, and indefinite integral (antiderivative) is necessary in understanding the fundamental theorem of calculus. The derivative can be thought of as measuring the change of the value of a variable with respect to another variable. The definite integral is the net area under the curve of a function and above the x-axis over an interval *[a,b]*. The indefinite integral (antiderivative) of a function *f* is another function *F* whose derivative is equal to the first function *f*.

### First Fundamental Theorem of Calculus

The first fundamental theorem of calculus states that if the function f(x) is continuous, then

This means that the definite integral over an interval *[a,b]* is equal to the antiderivative evaluated at b minus the antiderivative evaluated at a. This gives the relationship between the definite integral and the indefinite integral (antiderivative).

### Second Fundamental Theorem of Calculus

The second fundamental theorem of calculus states that if the function f is continuous, then

This means that the derivative of the integral of a function *f* with respect to the variable t over the interval *[a,x]* is equal to the function *f* with respect to *x*. This describes the derivative and integral as inverse processes. ^{2}

## Why

See Theoretical Knowledge Vs Practical Application.

## How

Many of the **References** and **Additional Reading** websites and **Videos** will assist you with TBD.

As some professors say: “It is intuitively obvious to even the most casual observer.“

## References

^{1} “Fundamental Theorem Of Calculus – Part 1, Part 2| Remarks & Examples”. 2022. *BYJUS*. https://byjus.com/maths/fundamental-theorem-calculus/.

^{2} “Fundamental Theorem Of Calculus – Simple English Wikipedia, The Free Encyclopedia”. 2022. *Wikipedia*. https://simple.wikipedia.org/wiki/Fundamental_theorem_of_calculus.

^{3} Yann Picand, Dominique Dutoit. 2022. “Fundamental Theorem Of Calculus : Definition Of Fundamental Theorem Of Calculus And Synonyms Of Fundamental Theorem Of Calculus (English)”. *dictionary.sensagent.com*. http://dictionary.sensagent.com/Fundamental%20theorem%20of%20calculus/en-en/.

## Additional Reading

Bocss, Manin. “The Fundamental Theorem Of Calculus”. 2022. *Medium*. https://maninbocss.medium.com/the-fundamental-theorem-of-calculus-30ac41807158.

⭐ “The Fundamental Theorem Of Calculus Made Clear: Intuition”. 2022. *Intuitive Calculus.Com*. http://www.intuitive-calculus.com/fundamental-theorem-of-calculus.html.

⭐ “Antiderivatives And The Fundamental Theorem Of Calculus | Khan Academy”. 2022. *Khan Academy*. https://www.khanacademy.org/math/old-ap-calculus-ab/ab-antiderivatives-ftc.

## Videos

This math video tutorial provides a basic introduction into the fundamental theorem of calculus part 1. It explains how to evaluate the derivative of the definite integral of a function f(t) using a simple process. f(x) is a continuous function on the closed interval [a, b] and F(x) is the antiderivative of f(x). You need to be familiar with the chain rule for derivatives. This video contains plenty of examples and practice problems.

This calculus video tutorial provides a basic introduction into the fundamental theorem of calculus part 2. It explains the process of evaluating a definite integral. F(x) is the antiderivative of f(x). This tutorial contains plenty of examples and practice problems.

This calculus video tutorial explains the concept of the fundamental theorem of calculus part 1 and part 2. This video contain plenty of examples and practice problems evaluating the definite integral using part 2 of FTC and finding the derivative of the integral of function. It discusses functions such as f(x) and F(x) the antiderivative of f(x) and shows the relationship between differentiation – the process of finding the derivative versus integration or antidifferentiation – the process of finding the integral of a function or its antiderivative.

This calculus video tutorial provides a basic introduction into the definite integral. It explains how to evaluate the definite integral of linear functions, rational functions, and those involving natural log functions. The indefinite integral gives you the antiderivative function in terms of x where as the definite integral gives you a specific number or constant as an answer. This video explains how to evaluate definite integrals using the fundamental theorem of calculus part 2. This tutorial contains plenty of examples and practice problems.

This calculus video tutorial explains how to find the indefinite integral of function. It explains how to apply basic integration rules and formulas to help you integrate functions. This video contains plenty of examples and practice problems.

This calculus video tutorial explains how to calculate the definite integral of function. It provides a basic introduction into the concept of integration. It provides plenty of examples and practice problems for you to work on.

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.