Introduction
The Fundamental Theorem 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 \( \displaystyle \int\limits_a^b f(x) \,dx = F(b) – F(a) \). 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. There is no one correct way; students should think about the Fundamental Theorem of Calculus 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 Theorem 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, namely \( \displaystyle \int\limits_a^b f(x) \,dx = F(b) – F(a) \). Consequently, the second FTC states that integrals and derivatives are inverses of each other. I Aced Calculus follows these names. Larson’s “Calculus” also follows this order; although Larson and the Art of Problem Solving textbooks instead of first FTC, just say FTC.
On the other hand, sources such as Khan Academy, Organic Chemistry Tutor, and Stewart switch this order. In reverse, they refer to the FTC part 1 as the theorem stating that integrals and derivatives are inverses. Consequently, they name the FTC part 2 the one that provides a method to calculate definite integrals by using antiderivatives. Therefore, there is a controversy about both the exact names and the order in which 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 and not splitting it into parts.
Who’s Right?
There is no consensus that dictates what the names and order of the Fundamental Theorem of Calculus should be. Many textbooks and resources have different preferences. Either order is correct because it is only a matter of presentation by the author. It is important to note that the content of the statements remains consistent. Thus, the importance of these theorems lies in their content rather than the exact names that they hold. Specifically, students should focus on understanding what the Fundamental Theorem of Calculus implies and how to apply the concepts to a variety of different calculus problems. Ultimately, while there isn’t an exact consensus on the naming of the Fundamental Theorem of Calculus, the content and utility of these theorems remain the same.
