The first fundamental theorem of calculus links differentiation and integration, two core concepts of calculus. In this article, we will explore the intricacies of the first fundamental theorem of calculus, breaking down its components, implications, and applications.
It is possible that your teacher or textbook call this theorem “The second fundamental theorem of calculus”, as there is no consensus – we have addressed this in detail in this article: The Second Fundamental Theorem of Calculus.
What is the First Fundamental Theorem of Calculus?
The first fundamental theorem of calculus establishes a direct link between differentiation and integration, two core operations in calculus. Specifically, it states that if \(f(x)\) is a continuous function on the closed interval \([a,b]\) and for \(x\) in \([a,b]\) we define \(F(x)\) as
\[F(x)=\int_a^x f(t) \,dt\]
then the derivative of \(F(x)\) with respect to \(x\) is equal to \(f(x)\):
\[F′(x)=f(x)\]
We may interpret this theorem as saying that the process of integration (finding the area under a curve) and differentiation (finding the slope of the tangent line of a curve) are inverse operations. This profound connection simplifies the process of solving various real-world problems involving rates of change and accumulated quantities.
Historical Background
The First Fundamental Theorem of Calculus was independently formulated by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Both scientists are credited for the discovery and reached the conclusion through wildly different methods. Newton approached calculus through his method of fluxions, while Leibniz developed a systematic notation that is still in use today. Despite their different methodologies, both mathematicians recognized the intrinsic relationship between differentiation and integration.
Breaking Down the Theorem
To fully grasp the First Fundamental Theorem of Calculus, it is essential to understand its components:
- Function Continuity: The function \(f(x)\) must be continuous on the closed interval \([a,b]\). Continuity ensures that there are no abrupt jumps or breaks in the function, allowing for a smooth accumulation of area under the curve.
- Integral Definition: The function \(F(x)\) is defined as the definite integral of \(f(t)\) from \(a\) to \(x\). This integral is equal to the area under the graph of \(f(t)\) from \(t=a\) to \(t=x\).
- Differentiation of the Integral: The theorem states that the derivative of \(F(x)\) with respect to \(x\) is equal to the integrand \(f(x)\). This means that the rate of change of the accumulated area under the curve is given by the height of the curve at each point.
An Easy Example
Let’s illustrate the First Fundamental Theorem of Calculus with an example:
Suppose we are given a function \(F(x)\) defined as the integral of \(f(t)=2t\) from \(t=1\):
\[F(x)=\int_1^x 2t \,dt\]
Evaluate the Integral: First, we need to find \(F(x)\) by integrating \(2t\) with respect to \(t\). For this, we will use the Second Fundamental Theorem of Calculus:
\[F(x)=\int_1^x2t \,dt=[t^2]_1^x=x^2−1\]
Differentiate \(F(x)\): Next, we differentiate \(F(x)\) with respect to \(x\):
\[F′(x) = \frac{d}{dx}(x^2 – 1) = 2x.\]
Thus, we have shown that \(F′(x)=2x\), which is the original function \(f(x)=2x\). This demonstrates that the derivative of the integral of a function returns the original function, as stated by the First Fundamental Theorem of Calculus.
Visualizing the Theorem
Let’s try to imagine the theorem applied to the special case above with \(f(t) = 2t\). Imagine plotting the function \(f(t)=2t\) on a graph and shading the area under the curve between \(t=1\) and \(t=x\). This shaded region represents the integral \(\int_1^x 2t dt\). By computing the integral, we find the accumulated area. If \(x\) increases by a tiny amount, let’s say from \(x\) to \(x+\varepsilon\), the area under the curve will increase approximately by \(\varepsilon\cdot 2x\) (base of a rectangle times the height of a rectangle, since the blue shape is roughly a rectangle). As we let \(\varepsilon\) get smaller and smaller, the rate of change of the area under the integral can be viewed as \(2x\), the value of the function. Note that \(\varepsilon\) gets closer to 0, but it is never equal to it. Otherwise, there would be no change in area.
Common Misconceptions
There are two primary misconceptions that can arise when learning about the First Fundamental Theorem of calculus:
1. Changes in the Bounds: If in the expression \(\displaystyle F(x) = \int_a^x f(t)\,dt\) the upper bound with the variable is modified, for example, \(\displaystyle F(x) = \int_a^{3x} f(t)\,dt\), we can no longer apply the First FTC the same way. We now have to use the Chain Rule:
\[F’(x) = f(3x)\cdot 3.\]
If the variable is the lower bound and not the upper bound, then the derivative becomes negative:
\[F(x) = \int_{x^2}^b f(t)\,dt \Rightarrow F’(x) = -f(x^2)\cdot 2x.\]
2. Function Continuity: The requirement for \(f(x)\) to be continuous on \([a,b]\) is crucial. If \(f(x)\) has discontinuities within the interval, the theorem does not apply, even though it may be modified slightly to account for those discontinuities.
Advanced Topics
For those interested in exploring further, there are many topics that the First Fundamental Theorem of Calculus is important in. Some examples are:
- Second Fundamental Theorem of Calculus: This theorem states that if \(f(x)\) is continuous on \([a,b]\), and we define a function \(F(x)\) such that \(\dfrac{d}{dx}(F(x)) = f(x)\), then \(\int_a^b f(x) \,dx = F(b) – F(a)\). The following blog will explain the Second Fundamental Theorem of calculus in greater detail: The Second Fundamental Theorem of Calculus
- Improper Integrals: When dealing with unbounded intervals or functions with discontinuities, improper integrals extend the concept of definite integrals. Techniques such as limits and comparison tests are used to evaluate these integrals.
- Numerical Integration: In cases where finding an exact antiderivative is challenging, numerical methods like the trapezoidal rule provide approximate solutions to definite integrals.
Conclusion
The first fundamental theorem of calculus is a pivotal concept that bridges the gap between differentiation and integration. By understanding its principles and applications, you can unlock a deeper comprehension of mathematical analysis and its real-world implications. Whether you’re solving complex engineering problems or analyzing economic trends, the theorem provides a powerful tool for tackling a wide range of challenges. Embrace this foundational theorem, and let it guide you through the fascinating world of calculus.
