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# The Fundamental Theorem of Calculus

Calculus is a branch of mathematics that deals with the study of rates of change and their relationship with functions. The Fundamental Theorem of Calculus is a theorem that connects the two main branches of calculus, namely differential calculus and integral calculus, by providing a method to evaluate definite integrals. It is often considered as one of the most important theorems in calculus. In this article, we will take a closer look at the Fundamental Theorem of Calculus, its statement, and how it can be applied.

## Statement of the Theorem

The Fundamental Theorem of Calculus has two parts, which are known as the first and second parts respectively.

### First Part

The first part of the theorem relates differentiation and integration. It states that if a function $f(x)$ is continuous on the closed interval $[a,b]$ and $F(x)$ is any antiderivative of $f(x)$, then:

$$
\int_a^b f(x) dx = F(b) - F(a)
$$

In other words, the definite integral of a function $f(x)$ over the interval $[a,b]$ is equal to the difference between the values of its antiderivative $F(x)$ at the endpoints $b$ and $a$. This part of the theorem is also known as the evaluation theorem.

### Second Part

The second part of the theorem gives a method for finding the derivative of an integral. It states that if the function $f(x)$ is continuous on the open interval $(a,b)$ and $F(x)$ is any antiderivative of $f(x)$, then:

$$
\frac{d}{dx} \int_a^x f(t) dt = f(x)
$$

In other words, the derivative of the integral of a function $f(x)$ with respect to its upper limit $x$ is equal to the function $f(x)$ itself. This part of the theorem is also known as the differentiation theorem.

## Applications of the Theorem

The Fundamental Theorem of Calculus has many applications in various fields of science and engineering. Here are some examples:

### Area under a Curve

One of the most common applications of the theorem is in finding the area under a curve. If we have a continuous function $f(x)$ on the interval $[a,b]$, we can find the area under the curve using the definite integral:

$$
\text{Area} = \int_a^b f(x) dx
$$

### Velocity and Acceleration

The theorem also has applications in physics, particularly in the study of motion. If we have a function that represents the velocity of an object, we can find the distance that the object has traveled by integrating the velocity function over time. The derivative of the velocity function gives the acceleration of the object.

### Probability Density Functions

The theorem is also used in probability theory to find the probability density function of a continuous random variable. The probability density function represents the probability of an event occurring within a certain range of values. It can be found by integrating the probability distribution function over the range of values.

## Conclusion

The Fundamental Theorem of Calculus is a powerful tool that connects differentiation and integration, and has many applications in various fields of science and engineering. It provides a method to evaluate definite integrals and find the derivative of an integral. The theorem is an essential concept in calculus and is often used in advanced mathematics and physics.
## Links
[微積分学の基本定理[JA]](/6455821d6729acbd37c9aa83)
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