What Is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a powerful tool that makes solving integrals and derivatives much simpler and faster than ever before. From rocket launches to economics, its applications are vast and varied, giving us an invaluable resource for solving complex problems quickly and accurately.

In this article, we'll explain the principle in detail as well as look at how it can be applied in practice.

Why Bother Learning the Fundamental Theorem of Calculus?

While it certainly has an intimidating name, the Fundamental Theorem of Calculus is not nearly as complicated as it seems. In fact, it's a fairly straightforward concept that can simplify your life.

The First Fundamental Theorem of Calculus is an essential concept to master for the simplicity and efficiency it offers in solving problems related to integration. It gives us a way to find definite integrals without having to calculate areas or use Riemann sums. This means that we can directly compute integrals by simply finding the derivative of a function and plugging in the necessary values.

The First Fundamental Theorem of Calculus is closely related to the Second Fundamental Theorem of Calculus, which evaluates a definite integral in terms of an antiderivative of its integrand. The two theorems together form the foundation of calculus, enabling us to apply calculus principles to a variety of scientific, engineering, and mathematical applications.

The work of developing the fundamental theorems of calculus is attributed to James Gregory, who published a rudimentary version of them in the 1600s. Since then, these theorems have been refined and expanded upon by mathematicians in the years that followed.

What is the Fundamental Theorem of Calculus? - How It Works

Not confused yet? Great! Let's dive deeper into the Fundamental Theorem of Calculus by explaining its parts and fundamentals in greater detail.

Derivatives and Integrals

To understand how this theorem works, we need to understand its two basic components: derivatives and integrals. A derivative is a measure of how much a function changes over a specified interval, while an integral is a measure of the area under a curve.

At its core, the Fundamental Theorem of Calculus is a way of connecting these concepts. It provides a bridge between the two, allowing us to calculate integrals and derivatives faster and with more accuracy.

The Principle Itself

Now it's time for the actual theorem. The Fundamental Theorem of Calculus establishes the relationship between a derivative and an integral, where the derivative of a function is equal to the integral of that same function.

Mathematically, this relationship can be expressed as: F(a) ∫abf(x)dx=F(b)−F(a)

This equation states that the integral of a function between two points (a and b) is equal to the change in the value of that function's antiderivative (F(b)−F(a)). We can use this equation to quickly and easily find the integral of a function without having to calculate the area manually.

The Theorem In Action

Say we want to calculate the area under a curve from x=1 to x=3. Following the Fundamental Theorem of Calculus, we'd start by finding the antiderivative of the function, which is F(x). We'd then plug in our initial and final values for x (1 and 3, respectively) to find the difference between them. This would give us the area under the curve from 1 to 3.

The Fundamental Theorem of Calculus in Practice

What would you actually use the Fundamental Theorem of Calculus for in real life? There are actually plenty of uses.

For instance, it can be used to calculate the velocity of an object over a given period of time. A rocket launching off, for example, would have an acceleration that could be expressed as a function of time. If you wanted to figure out the rocket's velocity between 5 and 10 seconds, you could use integration and the Fundamental Theorem of Calculus to calculate it quickly and easily.

The Fundamental Theorem of Calculus also makes it possible to solve complex physics problems, such as calculating the time it takes for a particle to reach a certain velocity. You could use the theorem to find the distance traveled by a particle over an interval of time or calculate the average velocity of a body throughout its motion.

Some areas of chemistry and engineering also use the Fundamental Theorem of Calculus. It can be used to find out how much heat an object has lost (or gained) over a certain interval, or to work out how much energy is required for a device to reach its desired state. You might even use it to calculate the rate of change in population or economic growth.

Final Thoughts

The Fundamental Theorem of Calculus is a powerful tool in mathematics that helps us understand the relationship between functions and their derivatives. Although its name is certainly a mouthful, the concept itself is relatively simple. 

By understanding the theorem and its applications, we can gain insights into how different areas of mathematics are connected and use it to solve more complex equations with greater accuracy. If you're looking to gain a better understanding of calculus, the Fundamental Theorem of Calculus is an excellent starting point. Calculus is a branch of mathematics that deals with the study of change and motion. Working with an online calculus tutor can help you quickly increase your knowledge.