The Fundamental Theorem of Calculus (FTC) is a pivotal concept in mathematics that connects the fields of differential calculus and integral calculus. This theorem is essential for students to understand, as it lays the groundwork for many advanced topics in calculus and analysis. In this article, we will explore the FTC in detail, breaking it down into its two main parts, providing examples, and discussing its significance in mathematics.

Introduction to the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus consists of two parts:

1. First Part: This part establishes the relationship between differentiation and integration.
2. Second Part: This part provides a method for evaluating definite integrals.

Understanding both parts is crucial for grasping how calculus functions as a whole.

Part 1: The First Fundamental Theorem of Calculus

The first part of the FTC states that if $f$ is a continuous function on the interval $[a,b]$ and $F$ is an antiderivative of $f$ on that interval, then:

∫abf(x) dx=F(b)−F(a)∫ab​f(x)dx=F(b)−F(a)

Explanation

• Continuous Function: A function is continuous if there are no breaks, jumps, or holes in its graph.
• Antiderivative: An antiderivative of a function $f$ is another function $F$ such that F′=fF′=f.

Example

Consider the function $f(x)=2x$. An antiderivative of $f$ is F(x)=x2F(x)=x2.

Using the first part of the FTC, we can evaluate the definite integral:

∫132x dx=F(3)−F(1)=32−12=9−1=8∫13​2xdx=F(3)−F(1)=32−12=9−1=8

Part 2: The Second Fundamental Theorem of Calculus

The second part of the FTC states that if $f$ is a continuous function on the interval $[a,b]$, then the function $F$ defined by:

F(x)=∫axf(t) dtF(x)=∫ax​f(t)dt

is continuous on $[a,b]$, differentiable on $(a,b)$, and F′(x)=f(x)F′(x)=f(x).

Explanation

• This part emphasizes that differentiation and integration are inverse processes.
• It shows that the process of integration can produce a new function whose derivative is the original function.

Example

Let’s consider the function $f(x)=\sin (x)$. We can define:

F(x)=∫0xsin⁡(t) dtF(x)=∫0x​sin(t)dt

According to the second part of the FTC, we find:

F′(x)=sin⁡(x)F′(x)=sin(x)

This confirms that $F(x)$ is indeed an antiderivative of $f(x)$.

Applications of the Fundamental Theorem of Calculus

The FTC has numerous applications in various fields, including:

• Physics: Calculating the area under curves, which can represent quantities like distance and work.
• Economics: Finding consumer and producer surplus through definite integrals.
• Engineering: Solving problems involving rates of change and accumulation.

Conclusion

The Fundamental Theorem of Calculus is a cornerstone of calculus that bridges the gap between differentiation and integration. By understanding both parts of the theorem, students can appreciate the powerful relationship between these two fundamental operations in mathematics.

For a more visual explanation and further examples, check out my YouTube video here: Fundamental Theorem of Calculus.