Fundamental Theorem of Calculus: which is first and which is second?


The Fundamental Theorems of Calculus (FTC) is extremely important for relating the study of derivatives and integrals. One part states that given a function \( f(x) \) and \( \displaystyle F(x) = \int_a^x f(t)\,dt \) it is true that \( \displaystyle F’(x) = f(x) \). \( F(x) \) can be viewed as an area accumulation function under the graph of \( f(x) \) and \( F’(x) \) as the rate of change of the area under the graph of \( f(x) \). This part of the theorem implies that derivatives and integrals are inverse functions of each other.

The other part states that, for any function \( f(x) \) and its antiderivative \( F(x) \), it is true that \( \int\limits_a^b f(x) \,dx = F(b) – F(a) \). This theorem states that the area under a curve on a certain interval can be found as a difference of the antiderivative evaluated at the two endpoints of that interval. Without this theorem, we would have to compute areas under curves by approximating them with rectangles (or other shapes), which would be very difficult and cumbersome. For example, \( \displaystyle\int_a^b \sin{x}\, dx \) represents the area under the graph of \( \sin x \) between the bounds \( a \) and \( b \), which can easily be calculated by finding the antiderivative of \( \sin x \). Without this theorem, finding the area under \( y = \sin x \) would be prohibitively difficult.

However, there is no consensus regarding which of these parts should come first (FTC 1) and which should come second (FTC 2). Additionally, some authors combine both parts into one theorem and call it the Fundamental Theorem of Calculus. This could be a source of confusion for the students who study from multiple sources. In short, there is no one correct way; students should think about these theorems as one big concept that different authors present in different ways, some choosing to partition that information into two parts. This article will cover how different sources present the theorem.

Discrepancies Between Sources

Different sources have different preferences regarding how to name the Fundamental Theorems of Calculus. For instance, CollegeBoard, Kaplan’s “Advanced Calculus” and Strauss’ “Calculus” state that the first FTC provides a method of evaluating definite integrals using antiderivatives. As a result, the second FTC would state that integrals and derivatives are inverses of each other. Larson’s “Calculus” also follows this order; although Larson and the Art of Problem Solving textbooks choose to name the first theorem as just FTC rather than FTC 1.

On the other hand, sources such as Khan Academy, Organic Chemistry Tutor, and Stewart switch this order. Instead, they refer to the first FTC as the theorem stating that integrals and derivatives are inverses. Consequently, they believe the second FTC provides a method to calculate definite integrals by using antiderivatives. Unlike Larson, they also number the theorems as FTC 1 and FTC 2, instead of using just FTC. Therefore, there is also controversy not only based on the names of the theorems, but also what order they go in.

Some sources, like Barron’s, decide to be completely neutral in the naming and ordering of the theorems by calling both theorems FTC.

Who’s Right?

As of now, there is no consensus that dictates what the naming and ordering of the Fundamental Theorems of Calculus should be. Many textbooks and resources have different preferences, making it impossible to label the theorems in a satisfactory way. As a result, either order is correct, because it is just a matter of preference for the author. It is important to note that the content of the statements remains consistent. Thus, the importance of these theorems lies with their content rather than what they are called. Specifically, students should focus on understanding what these theorems imply and how to apply their statements to a variety of different calculus problems. Ultimately, while there isn’t an exact consensus on the naming of the Fundamental Theorems of Calculus, the content and utility of these theorems remain the same.