# Fundamental Theorem of Calculus concept

• Wm_Davies
In summary, the fundamental theorem of calculus ties together differentiation and integration and makes it easier to solve definite integrals. This theorem can be applied to any function as long as its antiderivative can be found. This makes it a powerful tool in solving a wide range of integration problems.
Wm_Davies
I just learned about the fundamental theorem of calculus. I can see that this ties together differentiation and intergration, but I was wondering what kind of problems can be solved by using this theorem? In other words, what can the theorem be applied to?

The definite integral of a function f(x) is a number representing the area under the curve of f. The way it is defined is as a limiting process in which we approximate the area under the curve of f as the sum of the area of little rectangles. Most likely, you have computed the integral of a few functions using this definition. Those were pretty simple functions I'm sure... the likes of polynomials or "step functions". But for most functions it is hard to compute integrals using the definition alone.

And this is where the fundamental theorem of calculus comes into play! It says, well computing the integral of f(x) between a and b is easy: you only have to find a function F(x) such that F'(x) =f(x). Then
$$\int_a^bf(x)dx = F(b)-F(a)$$

Voila!

## 1. What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a fundamental concept in calculus that establishes a connection between the two major branches of calculus: differential calculus and integral calculus. It states that differentiation and integration are inverse operations, meaning that the derivative of a function is equal to the original function when integrated.

## 2. What is the significance of the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is significant because it allows for the evaluation of definite integrals without having to use Riemann sums or other complicated methods. It also provides a way to easily find the antiderivative of a function, which is important in many areas of mathematics and science.

## 3. How do you use the Fundamental Theorem of Calculus?

To use the Fundamental Theorem of Calculus, you first need to find the antiderivative of the function you are integrating. Then, you can simply substitute the upper and lower limits of integration into the antiderivative and subtract the two values to find the definite integral.

## 4. What is the difference between the first and second parts of the Fundamental Theorem of Calculus?

The first part of the Fundamental Theorem of Calculus states that if a function is continuous on a closed interval, then the definite integral of that function can be evaluated by finding its antiderivative and evaluating it at the upper and lower limits of integration. The second part of the theorem states that if a function is continuous on a closed interval, then the derivative of the definite integral of that function is equal to the original function.

## 5. Can the Fundamental Theorem of Calculus be applied to all functions?

No, the Fundamental Theorem of Calculus can only be applied to certain types of functions, specifically those that are continuous on a closed interval. This means that the function must not have any "jumps" or discontinuities within the interval. Additionally, the function must have a well-defined antiderivative, which is not always the case for all functions.

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