Proving The Fundamental Theorem Of Calculus

Proving the Fundamental Theorem of Calculus: A Deep Dive

The Fundamental Theorem of Calculus is a cornerstone of integral calculus, elegantly linking differentiation and integration. It's not just a theorem; it's a bridge, connecting two seemingly disparate branches of calculus into a unified and powerful whole. This theorem allows us to calculate definite integrals using antiderivatives, significantly simplifying the process of calculating areas under curves. However, simply knowing the theorem isn't enough; understanding its proof is crucial to truly grasping its power and implications. This article delves deep into the intricacies of proving both parts of the Fundamental Theorem of Calculus, providing a detailed explanation for both the first and second parts.

Understanding the Two Parts of the Fundamental Theorem of Calculus

Before we embark on the proofs, let's refresh our understanding of the theorem itself. It's comprised of two main parts:

Part 1 (The First Fundamental Theorem of Calculus): This part establishes the connection between differentiation and integration. It states that if F(x) is an antiderivative of f(x) on an interval [a, b], then:

∫_a^b f(x) dx = F(b) - F(a)

This means that the definite integral of a function can be evaluated by finding an antiderivative and subtracting its values at the limits of integration.

Part 2 (The Second Fundamental Theorem of Calculus): This part deals with the derivative of an integral. It states that if f(x) is continuous on an interval containing a, then:

d/dx ∫_a^x f(t) dt = f(x)

This means that the derivative of the integral of a function, with respect to its upper limit, is simply the original function.

Proving the First Fundamental Theorem of Calculus

The proof of the first fundamental theorem relies heavily on the Mean Value Theorem. Let's break down the steps:

1. Defining the Problem:

We start with a function f(x) that is continuous on the closed interval [a, b], and we assume that F(x) is an antiderivative of f(x). This means that F'(x) = f(x) for all x in [a, b].

2. Introducing the Mean Value Theorem:

The Mean Value Theorem states that for a continuous function on a closed interval and differentiable on the open interval, there exists at least one point c within the interval where the instantaneous rate of change equals the average rate of change. Mathematically:

F'(c) = (F(b) - F(a)) / (b - a)

3. Leveraging the Antiderivative:

Since F'(x) = f(x), we can substitute f(x) for F'(c):

f(c) = (F(b) - F(a)) / (b - a)

4. Introducing the Riemann Sum:

Now, let's consider the definite integral of f(x) from a to b. We can approximate this integral using a Riemann sum:

∫_a^b f(x) dx ≈ Σ_i=1ⁿ f(x_i^*)Δx

where Δx = (b - a) / n and x_i^* is a point in the i-th subinterval.

5. Connecting the Riemann Sum to the Mean Value Theorem:

As the number of subintervals n approaches infinity (making Δx infinitesimally small), the Riemann sum converges to the definite integral. If we choose x_i = c_i, where c_i is a point in each subinterval satisfying the Mean Value Theorem, we can rewrite the Riemann sum as:

Σ_i=1ⁿ f(c_i)Δx = Σ_i=1ⁿ [(F(x_i) - F(x_i-1))] = F(b) - F(a)

6. Conclusion:

As n approaches infinity, the Riemann sum approaches the definite integral. Therefore, we arrive at the conclusion of the First Fundamental Theorem:

∫_a^b f(x) dx = F(b) - F(a)

This proof elegantly demonstrates the fundamental connection between integration (represented by the definite integral) and differentiation (represented by the antiderivative F(x)).

Proving the Second Fundamental Theorem of Calculus

The proof of the second fundamental theorem relies on the definition of the derivative and properties of limits.

1. Defining the Problem:

Let G(x) = ∫_a^x f(t) dt, where f(t) is continuous on an interval containing a. Our goal is to show that G'(x) = f(x).

2. Utilizing the Definition of the Derivative:

The derivative of G(x) is defined as:

G'(x) = lim_h→0 [(G(x + h) - G(x)) / h]

3. Substituting the Integral Definition of G(x):

Substituting the integral definition of G(x), we get:

G'(x) = lim_h→0 [∫_a^x+h f(t) dt - ∫_a^x f(t) dt] / h

4. Utilizing Properties of Integrals:

We can combine the two integrals:

G'(x) = lim_h→0 [∫_x^x+h f(t) dt] / h

5. Applying the Mean Value Theorem for Integrals:

The Mean Value Theorem for Integrals states that there exists a number c in the interval [x, x+h] such that:

∫_x^x+h f(t) dt = f(c)(x + h - x) = f(c)h

6. Substituting and Simplifying:

Substituting this into the expression for G'(x):

G'(x) = lim_h→0 [f(c)h] / h = lim_h→0 f(c)

7. Utilizing Continuity:

Since f(t) is continuous, as h approaches 0, c approaches x. Therefore:

lim_h→0 f(c) = f(x)

8. Conclusion:

Therefore, we conclude that:

G'(x) = f(x)

This completes the proof of the second fundamental theorem. It shows that differentiation and integration are inverse operations, reinforcing the fundamental connection established by the first part of the theorem.

Applications and Significance of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is not just a theoretical result; it's a powerful tool with far-reaching applications across various fields:

• Physics: Calculating velocity from acceleration, displacement from velocity, and work done by a force are all applications that leverage this theorem.

• Engineering: Determining areas and volumes of irregular shapes, essential for structural analysis and design, relies on integration made practical by this theorem.

• Economics: Analyzing marginal cost and revenue, and calculating total cost and revenue from marginal functions uses integral calculus facilitated by the fundamental theorem.

• Probability and Statistics: Calculating probabilities and expected values often involves integration, made significantly simpler by the fundamental theorem.

• Computer Science: Numerical methods for solving differential equations and evaluating integrals rely heavily on the principles established by this theorem.

The Fundamental Theorem of Calculus, with its two interconnected parts, forms the foundation of much of modern calculus and its numerous applications. Understanding its proof, as detailed above, provides a deeper appreciation for the elegance and power of this fundamental concept in mathematics. The theorem bridges the gap between the seemingly distinct worlds of differentiation and integration, ultimately simplifying complex calculations and providing a crucial tool for solving problems in diverse fields. Mastering the proof is a significant step towards true mastery of calculus.