8.1.1 Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is one of the most important results in calculus, linking the concepts of differentiation and integration. It establishes a direct connection between these two fundamental operations and provides a way to evaluate definite integrals.

1. Statement of the Fundamental Theorem of Calculus:

The Fundamental Theorem of Calculus has two parts:

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Part 1: The Integral as an Antiderivative (FTC1):

If $f(x)$ is continuous on a closed interval $[a, b]$ , then the function $\ F(x)$ defined by: $\ F(x) = \int_a^x f(t) \, dt$ is continuous on $\ [a, b]$ , differentiable on the open interval $\ (a, b),$ and its derivative is $\ f(x)$ . In other words: $F'(x) = \frac{d}{dx} \left(\int_a^x f(t) \, dt\right) = f(x)$

Interpretation: This part of the theorem tells us that if we integrate a function $\ f(x)$ from $\ a$ to $\ x$ , and then differentiate the result, we get back the original function $\ f(x)$ .

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Part 2: The Evaluation of Definite Integrals (FTC2):

If $\ f(x)$ is continuous on the closed interval $\ [a, b]$ and $F(x)$ is any antiderivative of $\ f(x) \ (i.e., \ F'(x) = f(x)$ ), then: $\\ \int_a^b f(x) \, dx = F(b) - F(a)$

Interpretation: This part of the theorem allows us to evaluate the definite integral of $\ f(x)$ over $[a, b]$ by finding any antiderivative $\ F(x) \ of \ f(x)$ , and then computing the difference $\ F(b) - F(a) .$

2. Understanding the Two Parts Together:

FTC1 provides the link between differentiation and integration, showing that integration (specifically the process of finding an accumulated area under a curve) can be "undone" by differentiation.
FTC2 gives us a powerful tool for evaluating definite integrals without needing to perform the limiting process of summing Riemann sums; instead, we can use antiderivatives directly.

3. Examples Illustrating the Fundamental Theorem of Calculus:

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Example 1: Applying FTC1

Let $\ f(x) = 2x ,$ and define $\ F(x)$ as: $F(x) = \int_0^x 2t \, dt$

To find $\ F(x)$ , we integrate: $F(x) = \left. t^2 \right|_0^x = x^2$
Now, differentiate $\ F(x)$ to verify FTC1: $F'(x) = \frac{d}{dx}(x^2) = 2x$
As expected, $\ F'(x) = f(x)$ , confirming that the derivative of the integral gives back the original function.

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Example 2: Applying FTC2

Evaluate the definite integral: $\int_1^4 3x^2 \, dx$

First, find an antiderivative $\ F(x) \ of \ f(x) = 3x^2$ : $F(x) = x^3$
Now, apply FTC2: $\int_1^4 3x^2 \, dx = F(4) - F(1) = 4^3 - 1^3 = 64 - 1 = 63$
So, the value of the definite integral is 63.

4. Visualizing the Fundamental Theorem of Calculus:

FTC1 Visualization: Imagine $\ F(x)$ as the accumulated area under the curve $\ y = f(t) \ from \ t = a \ to \ t = x \ . As \ x$ changes, $\ F(x)$ changes. The rate at which $\ F(x)$ changes (i.e., $\ F'(x)$ ) is given by the height of the curve at $\ x$ , which is $\ f(x)$ .
FTC2 Visualization: When evaluating $\ \int_a^b f(x) \ , dx$ , you're finding the net area between the curve $\ y = f(x)$ and the x-axis from $\ x = a \ to \ x = b$ . FTC2 allows you to find this area by computing the difference between the values of an antiderivative at $\ b \ and \ a$ .

Summary:

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The Fundamental Theorem of Calculus connects differentiation and integration, showing that they are essentially inverse processes.
FTC1 tells us that the derivative of the integral of a function is the function itself.
FTC2 allows us to evaluate definite integrals by finding an antiderivative and computing the difference in its values at the endpoints.
This theorem is central to the practice of calculus and has profound implications in many areas of mathematics, physics, engineering, and beyond.

Fundamental Theorem of Calculus Simplified Revision Notes for A-Level AQA Maths Pure

8.1.1 Fundamental Theorem of Calculus

1. Statement of the Fundamental Theorem of Calculus:

Part 1: The Integral as an Antiderivative (FTC1):

Part 2: The Evaluation of Definite Integrals (FTC2):

2. Understanding the Two Parts Together:

3. Examples Illustrating the Fundamental Theorem of Calculus:

Example 1: Applying FTC1

Example 2: Applying FTC2

4. Visualizing the Fundamental Theorem of Calculus:

Summary:

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