Proof Of Fundamental Theorem Of Calculus

The fundamental theorem of calculus bridges the seemingly disparate concepts of differentiation and integration, establishing a profound connection between the slope of a curve and the area under it. At its core, this theorem states that differentiation and integration are, in a sense, inverse operations.

Two Pillars: Part 1 and Part 2

The fundamental theorem of calculus is typically presented in two parts:

• Part 1: Deals with the derivative of an integral function. It states that if f is a continuous function on the interval [a, b] and F is defined by:

  F(x) = ∫ₐˣ f(t) dt

  Then F is differentiable on [a, b], and its derivative is f(x). In essence, this part demonstrates how to construct an antiderivative for a continuous function.

• Part 2: Provides a method for calculating definite integrals using antiderivatives. It asserts that if f is a continuous function on [a, b] and F is any antiderivative of f (meaning F'(x) = f(x)), then:

  ∫ₐᵇ f(x) dx = F(b) - F(a)

  This part offers a practical way to evaluate definite integrals, which are crucial for many applications in science and engineering.

Why is it "Fundamental"?

The theorem's importance stems from several factors:

1. It links two major branches of calculus: Differential calculus (dealing with rates of change) and integral calculus (dealing with accumulation).
2. It simplifies the computation of definite integrals: Before the theorem, evaluating definite integrals often required cumbersome limit processes.
3. It guarantees the existence of antiderivatives: Part 1 shows that every continuous function has an antiderivative, although finding a closed-form expression for it might be challenging.

Proving Part 1: The Derivative of an Integral Function

To rigorously prove Part 1, we need to show that the limit definition of the derivative of F(x) yields f(x).

1. Start with the Definition of the Derivative:

The derivative of F(x) is defined as:

F'(x) = lim(h→0) [F(x + h) - F(x)] / h

2. Substitute the Integral Definition of F(x):

Substitute the definition of F(x) into the limit:

F'(x) = lim(h→0) [∫ₐˣ⁺ʰ f(t) dt - ∫ₐˣ f(t) dt] / h

3. Combine the Integrals:

Using the properties of definite integrals, we can combine the two integrals into a single integral:

F'(x) = lim(h→0) [∫ₓˣ⁺ʰ f(t) dt] / h

4. Apply the Mean Value Theorem for Integrals:

Since f is continuous on [a, b], it is also continuous on [x, x + h]. Therefore, we can apply the Mean Value Theorem for Integrals, which states that there exists a number c in [x, x + h] such that:

∫ₓˣ⁺ʰ f(t) dt = f(c) * (x + h - x) = f(c) * h

Substitute this into the limit:

F'(x) = lim(h→0) [f(c) * h] / h

5. Simplify and Evaluate the Limit:

The h terms cancel out:

F'(x) = lim(h→0) f(c)

As h approaches 0, x + h approaches x. Since c is trapped between x and x + h, c must also approach x. Because f is continuous:

lim(h→0) f(c) = f(x)

Therefore:

F'(x) = f(x)

This completes the proof of Part 1.

Proving Part 2: Evaluating Definite Integrals

To prove Part 2, we need to show that if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) - F(a).

1. Define a New Function G(x):

Let's define a function G(x) as:

G(x) = ∫ₐˣ f(t) dt

2. Apply Part 1:

From Part 1, we know that G'(x) = f(x).

3. Relate G(x) to F(x):

We are given that F(x) is also an antiderivative of f(x), meaning F'(x) = f(x). Since both F(x) and G(x) have the same derivative, they must differ by a constant. Therefore, there exists a constant C such that:

F(x) = G(x) + C

4. Determine the Value of the Constant C:

To find C, let's evaluate F(a):

F(a) = G(a) + C

By the definition of G(x):

G(a) = ∫ₐᵃ f(t) dt = 0

Therefore:

F(a) = 0 + C => C = F(a)

5. Substitute the Value of C:

Now we know that:

F(x) = G(x) + F(a)

6. Evaluate F(b):

Let's evaluate F(b):

F(b) = G(b) + F(a)

Again, by the definition of G(x):

G(b) = ∫ₐᵇ f(t) dt

Therefore:

F(b) = ∫ₐᵇ f(t) dt + F(a)

7. Rearrange the Equation:

Rearranging the equation, we get:

∫ₐᵇ f(t) dt = F(b) - F(a)

Since t is just a variable of integration, we can replace it with x:

∫ₐᵇ f(x) dx = F(b) - F(a)

This completes the proof of Part 2.

Illustrative Examples

Let's solidify our understanding with a couple of examples:

Example 1: Finding the Derivative of an Integral

Let F(x) = ∫₀ˣ t² dt. Find F'(x).

Using Part 1 of the fundamental theorem, we know that F'(x) = x². That's it! The theorem allows us to directly find the derivative without actually evaluating the integral.

Example 2: Evaluating a Definite Integral

Evaluate ∫₁³ 2x dx.

First, we need to find an antiderivative of f(x) = 2x. An antiderivative is F(x) = x².

Then, using Part 2 of the fundamental theorem:

∫₁³ 2x dx = F(3) - F(1) = 3² - 1² = 9 - 1 = 8

Therefore, ∫₁³ 2x dx = 8.

A More Rigorous Approach to Part 1 (Optional)

While the previous proof is widely accepted, a more rigorous approach avoids direct reliance on the Mean Value Theorem for Integrals and provides a more detailed analysis of the limit.

1. Start with the Definition of the Derivative (Same as Before):

F'(x) = lim(h→0) [F(x + h) - F(x)] / h

2. Substitute the Integral Definition of F(x) (Same as Before):

F'(x) = lim(h→0) [∫ₐˣ⁺ʰ f(t) dt - ∫ₐˣ f(t) dt] / h

3. Combine the Integrals (Same as Before):

F'(x) = lim(h→0) [∫ₓˣ⁺ʰ f(t) dt] / h

4. Use the Definition of Continuity (Key Difference):

Since f is continuous at x, for any ε > 0, there exists a δ > 0 such that if |t - x| < δ, then |f(t) - f(x)| < ε. In other words, we can make f(t) arbitrarily close to f(x) by making t sufficiently close to x.

5. Bound the Integral:

Assume 0 < h < δ. Then for all t in the interval [x, x + h], we have |t - x| ≤ h < δ. Therefore, |f(t) - f(x)| < ε, which means:

f(x) - ε < f(t) < f(x) + ε

Now we can bound the integral:

∫ₓˣ⁺ʰ [f(x) - ε] dt < ∫ₓˣ⁺ʰ f(t) dt < ∫ₓˣ⁺ʰ [f(x) + ε] dt

Evaluating the integrals on the left and right:

[f(x) - ε] * h < ∫ₓˣ⁺ʰ f(t) dt < [f(x) + ε] * h

6. Divide by h:

Dividing all parts of the inequality by h (since h > 0, the inequality signs don't change):

f(x) - ε < [∫ₓˣ⁺ʰ f(t) dt] / h < f(x) + ε

7. Apply the Squeeze Theorem:

We have shown that for any ε > 0, we can find a δ > 0 such that if 0 < h < δ, then:

f(x) - ε < [∫ₓˣ⁺ʰ f(t) dt] / h < f(x) + ε

This means that:

lim(h→0⁺) [∫ₓˣ⁺ʰ f(t) dt] / h = f(x)

A similar argument can be made for h approaching 0 from the left (h < 0). Therefore:

lim(h→0) [∫ₓˣ⁺ʰ f(t) dt] / h = f(x)

Which implies:

F'(x) = f(x)

This more rigorous proof relies directly on the definition of continuity and the squeeze theorem, providing a more solid foundation for the result. The key is to use the continuity of f to bound the integral and then use the squeeze theorem to evaluate the limit.

Common Misconceptions and Pitfalls

• Forgetting the Constant of Integration: When finding an antiderivative, remember that there are infinitely many antiderivatives that differ by a constant. While this constant cancels out in definite integrals (Part 2), it's crucial in indefinite integrals.
• Assuming All Functions Have Elementary Antiderivatives: Not all functions have antiderivatives that can be expressed in terms of elementary functions (polynomials, exponentials, trigonometric functions, etc.). For example, e^(-x²) has an antiderivative, but it cannot be expressed in a simple, closed form. This doesn't invalidate the fundamental theorem; it just means we can't easily find the antiderivative.
• Discontinuities: The fundamental theorem requires the function f to be continuous on the interval of integration. If f has a discontinuity, the theorem might not apply directly, and you might need to break the integral into smaller intervals where f is continuous.
• Incorrectly Applying Part 1: Part 1 gives the derivative of an integral with a variable upper limit of integration. If the upper limit is a constant, the derivative is zero. Also, if the upper limit is a function of x (e.g., ∫ₐ^(g(x)) f(t) dt), you need to use the chain rule in conjunction with Part 1.

Applications of the Fundamental Theorem

The fundamental theorem of calculus has far-reaching applications in various fields:

• Physics: Calculating displacement from velocity (integration) and velocity from acceleration (differentiation). Determining work done by a force.
• Engineering: Analyzing circuits, designing control systems, and modeling fluid flow.
• Economics: Calculating consumer surplus and producer surplus.
• Statistics: Defining probability distributions and calculating expected values.
• Computer Graphics: Rendering curves and surfaces.

In essence, any field that relies on modeling continuous change and accumulation makes extensive use of the fundamental theorem of calculus.

The Broader Significance

The fundamental theorem of calculus is more than just a computational tool; it represents a profound intellectual achievement. It reveals a deep and unexpected connection between two seemingly different mathematical concepts, providing a powerful framework for understanding and modeling the world around us. It showcases the elegance and interconnectedness of mathematics, solidifying its place as a cornerstone of modern science and technology. Understanding and appreciating this theorem is crucial for anyone seeking a deeper understanding of calculus and its applications. It empowers us to solve problems, make predictions, and gain insights into the behavior of complex systems.