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Exploring the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is one of the most profound results in mathematics, bridging the concepts of differentiation and integration. This post will explore the theorem in detail, providing intuition, formal statements, and examples.

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The Two Parts of the Fundamental Theorem of Calculus

Part 1: Differentiation of the Integral

The first part of the FTC states that if $f$ is a continuous function on an interval $[a, b]$, and we define a new function $F(x)$ as:

$$F(x) = \int_a^x f(t) \, dt,$$

then $F'(x) = f(x)$ for all $x \in [a, b]$. In other words, the derivative of the integral of $f$ gives back $f$ itself.

This result provides a deep connection between differentiation and integration. Let’s break this down:

1. Intuition: The integral accumulates the "area under the curve" of $f(t)$ from $a$ to $x$. Differentiating this accumulation with respect to $x$ gives back the instantaneous rate of change of $f$ at $x$, which is simply $f(x)$.

2. Key Assumption: The function $f$ must be continuous on the interval $[a, b]$ for this result to hold.

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Part 2: The Evaluation Theorem

The second part of the FTC states that if $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ (i.e., $F'(x) = f(x)$), then:

$$\int_a^b f(x) \, dx = F(b) - F(a).$$

This result simplifies the computation of definite integrals by allowing us to evaluate the antiderivative at the endpoints of the interval.

1. Intuition: Instead of summing up infinitesimal areas under the curve directly, we can simply find an antiderivative $F$ and compute the difference $F(b) - F(a)$.

2. Key Assumption: Again, the continuity of $f$ on $[a, b]$ is crucial.

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A Proof Sketch for the First Part

To justify the first part of the FTC, let’s consider:

$$F(x) = \int_a^x f(t) \, dt.$$

The derivative of $F(x)$ is given by the definition of the derivative:

$$F'(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h}.$$

Expanding $F(x+h)$ and $F(x)$ using the definition of $F$, we have:

$$F(x+h) - F(x) = \int_a^{x+h} f(t) \, dt - \int_a^x f(t) \, dt.$$

Using the additivity property of definite integrals:

$$\int_a^{x+h} f(t) \, dt = \int_a^x f(t) \, dt + \int_x^{x+h} f(t) \, dt,$$

we find that:

$$F(x+h) - F(x) = \int_x^{x+h} f(t) \, dt.$$

Dividing by $h$:

$$\frac{F(x+h) - F(x)}{h} = \frac{1}{h} \int_x^{x+h} f(t) \, dt.$$

As $h \to 0$, the interval $[x, x+h]$ shrinks, and since $f$ is continuous, the average value of $f$ on this interval converges to $f(x)$. Thus:

$$\lim_{h \to 0} \frac{1}{h} \int_x^{x+h} f(t) \, dt = f(x),$$

which implies:

$$F'(x) = f(x).$$

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Examples

Example 1: A Simple Integral

Let $f(x) = 2x$. Compute $\int_0^3 2x \, dx$ using the FTC.

1. Find an antiderivative: $F(x) = x^2$.
2. Apply the evaluation theorem:

$$\int_0^3 2x \, dx = F(3) - F(0) = 3^2 - 0^2 = 9.$$

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Example 2: Differentiating an Integral

Let $f(t) = \sin(t)$ and define $F(x) = \int_0^x \sin(t) \, dt$. Compute $F'(x)$.

By the first part of the FTC:

$$F'(x) = \sin(x).$$

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Applications and Insights

The FTC is essential in various fields:

1. Physics: Calculating work done, displacement, and energy often involves definite integrals.
2. Engineering: Analysis of systems, such as finding the response of circuits or mechanical systems, uses integrals.
3. Probability: The cumulative distribution function (CDF) of a random variable is an integral, and its derivative gives the probability density function (PDF).

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Conclusion

The Fundamental Theorem of Calculus is a cornerstone of mathematical analysis, providing a bridge between two seemingly distinct concepts: differentiation and integration. By understanding both parts of the theorem, one gains powerful tools to solve a wide range of problems in mathematics, science, and engineering.

If you’ve made it this far, try applying the FTC to your own examples! For instance:

• Compute $\int_1^4 (3x^2 - 2x + 1) \, dx$.
• Differentiate $G(x) = \int_2^x e^{-t^2} \, dt$.

Feel free to share your solutions and thoughts below!