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Chapter 5: Problem 3

Explain in words and write mathematically how the Fundamental Theorem of Calculus is used to evaluate definite integrals.

Short Answer

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Question: Explain the Fundamental Theorem of Calculus and demonstrate how it can be used to evaluate definite integrals. Answer: The Fundamental Theorem of Calculus connects differentiation and integration by stating that every continuous function has an antiderivative. It has two parts: Part 1 explains that the derivative of an antiderivative is equal to the original function, while Part 2 expresses evaluating definite integrals using the antiderivative of the function and its endpoints (F(b) - F(a)). To evaluate a definite integral, one must find the antiderivative of the function, evaluate it at the endpoints of the interval, and subtract the results. For example, when evaluating \int_0^3 2x dx, the antiderivative is x^2, and the result is (3^2 - 0^2) = 9.

Step by step solution

01

Definition of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a result that connects differentiation and integration, and it is one of the most important theorems in calculus. It states that every continuous function has an antiderivative, and it has two important parts: Part 1 and Part 2.
02

The First Part of the Fundamental Theorem of Calculus

Part 1 states that if a function is continuous on the closed interval [a, b], then the function has an antiderivative F on the interval, and the derivative of F is equal to the original function. Mathematically, this is represented as: F'(x) = f(x) Where F(x) is the antiderivative of the continuous function f(x).
03

The Second Part of the Fundamental Theorem of Calculus

Part 2 states that if a function is continuous on the closed interval [a, b], and F is any antiderivative of the function on the interval, then the definite integral of the function from a to b can be evaluated as follows: \int_a^b f(x)dx = F(b) - F(a) This means that the definite integral of a function from a to b is equal to the difference of the antiderivative of the function evaluated at the endpoints of the interval.
04

Connection between the derivatives and the integrals

The Fundamental Theorem of Calculus shows that the process of finding the antiderivative (integration) is essentially the reverse of finding the derivative (differentiation). Together, differentiation and integration are the two main operations in calculus, and the Fundamental Theorem of Calculus provides a powerful tool for connecting these two seemingly disparate concepts.
05

Example of Evaluating a Definite Integral Using the Fundamental Theorem of Calculus

Here is a simple example to illustrate how the theorem is used to evaluate definite integrals: Consider the function f(x) = 2x. The task is to compute the following definite integral: \int_0^3 2x dx Step 1: Find the antiderivative F(x) of the function f(x) = 2x. In this case, F(x) = x^2. Step 2: Apply the second part of the Fundamental Theorem of Calculus: \int_0^3 2x dx = F(3) - F(0) Step 3: Evaluate the antiderivative F(x) at the endpoints of the interval: F(3) = (3)^2 = 9 F(0) = (0)^2 = 0 Step 4: Substitute the values back into the expression obtained in Step 2: \int_0^3 2x dx = 9 - 0 = 9 Thus, using the Fundamental Theorem of Calculus, we have found the definite integral of the function f(x) = 2x on the interval [0, 3] to be equal to 9.

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