Question

Explain mathematically how the Fundamental Theorem of Calculus is used to evaluate definite integrals.

Short answer

Question: Explain mathematically how the Fundamental Theorem of Calculus is used to evaluate definite integrals. Answer: The Fundamental Theorem of Calculus connects differentiation and integration. To evaluate definite integrals using this theorem, we find an antiderivative of the given continuous function, then evaluate it at the endpoints of the interval. The difference of these values gives the value of the definite integral. In the example of \(\int_0^2 x^2 dx\), we determined the antiderivative as \(\frac{1}{3}x^3\) and evaluated it at the endpoints \(0\) and \(2\). The difference of these values, \(\frac{8}{3} - 0\), gave us the definite integral value, \(\frac{8}{3}\).

Step-by-step solution

Define the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is a crucial theorem that connects differentiation and integration. It has two parts: 1. The first part states that if a function is continuous over an interval, then the function's integral is differentiable over that interval, and the derivative of the integral gives back the original function. 2. The second part states that if a function is continuous over an interval and has an antiderivative, then the definite integral of the function over that interval can be computed using the antiderivative. For this exercise, we will focus on the second part, also known as the evaluation theorem, which can be written as: $$ \int_a^b f(x) dx = F(b) - F(a) $$ Where \(f(x)\) is a continuous function on the interval \([a, b]\), and \(F(x)\) is an antiderivative of \(f(x)\).

Explain the theorem

The evaluation theorem states that to find the definite integral of a continuous function over a specific interval, we only need to find an antiderivative of that function and evaluate it at the endpoints of the interval. The theorem allows us to compute definite integrals more efficiently, as we no longer need to compute the limit of a Riemann sum. In practice, to evaluate a definite integral using the Fundamental Theorem of Calculus, we follow these steps: 1. Find an antiderivative \(F(x)\) of the function \(f(x)\). 2. Evaluate \(F(x)\) at the endpoints \(b\) and \(a\) of the interval. 3. Subtract the values from step 2, i.e., compute \(F(b) - F(a)\).

Example of using the theorem to evaluate a definite integral

Let's demonstrate the use of the Fundamental Theorem of Calculus through an example: Find the definite integral: $$ \int_0^2 x^2 dx $$ Following the steps mentioned earlier: 1. Find an antiderivative of \(f(x) = x^2\). In this case, \(F(x) = \frac{1}{3}x^3\) (i.e., the power rule for integration). 2. Evaluate \(F(x)\) at the endpoints: $$ F(2) = \frac{1}{3}(2)^3 = \frac{8}{3} $$ $$ F(0) = \frac{1}{3}(0)^3 = 0 $$ 3. Subtract the values: $$ \int_0^2 x^2 dx = F(2) - F(0) = \frac{8}{3} - 0 = \frac{8}{3} $$ Thus, the value of the definite integral is \(\frac{8}{3}\).

Summary

The Fundamental Theorem of Calculus helps us evaluate definite integrals by using the antiderivatives of functions. By finding an antiderivative of a continuous function, we can evaluate the integral over a specific interval by calculating the difference of the antiderivative values at the endpoints of that interval. In the provided example, we used the theorem to find the value of the definite integral \(\int_0^2 x^2 dx\) as \(\frac{8}{3}\).

Key concepts

Definite Integrals

A definite integral represents the signed area under a curve described by a function over a specific interval on the x-axis. Imagine you are measuring the area under a curve between two points. This is essentially what a definite integral does. It gives you a precise numerical value that represents this area.

Definite integrals are mathematically expressed as:

• \( \int_a^b f(x) \, dx \)

Here, \( a \) and \( b \) are the boundaries of the interval, and \( f(x) \) is the function you're integrating. The numbers \( a \) and \( b \) are limits of integration.

To compute a definite integral, we use the Fundamental Theorem of Calculus, which connects integrals with antiderivatives. This theorem takes advantage of knowing the antiderivative to readily calculate the integral, without resorting to infinitely adding slices or solving limits directly. This way, you can find the accumulated total change of the function in a swift and efficient manner.

Antiderivative

The antiderivative of a function is another function whose derivative is the original function. In simpler terms, if you take the derivative of the antiderivative, you'll end up with the original function. It's essentially the reverse process of differentiation.

For example, if you have a function \( f(x) \), an antiderivative would be a function \( F(x) \) such that:

• \( F'(x) = f(x) \)

Finding this "reverse" function is central to solving integrals. Notably, different antiderivatives can vary by a constant, because derivatives of constants are zero.

Using antiderivatives to evaluate definite integrals is one of the hallmarks of the Fundamental Theorem of Calculus. Consider the integral \( \int_a^b f(x) \, dx \). Here, \( F(x) \), an antiderivative of \( f(x) \), is used to quickly find the integral by computing \( F(b) - F(a) \). This approach simplifies the process considerably, eliminating the more tedious method of evaluating limits.

Continuous Function

Continuous functions are those without any breaks, jumps, or interruptions over an interval. They can be drawn without lifting your pencil from the paper. When a function is called continuous on an interval \([a, b]\), it means that for every point between \(a\) and \(b\), the function has a predictable and seamless behavior.

Why is this important in calculus? Continuous functions ensure that the operations we perform, like differentiation and integration, behave nicely. They guarantee that the integral calculations, such as those using the Fundamental Theorem of Calculus, can be performed without unexpected results or undefined zones.

In the context of the Fundamental Theorem of Calculus, having a continuous function is crucial. The theorem specifically requires the function to be continuous over the interval of integration, \([a, b]\). If this condition is met, you can confidently use the theorem to find the definite integral as the difference between the antiderivative evaluated at the interval's endpoints. Being continuous ensures that there will be no mysterious jumps in value or behavior of the function that could otherwise disrupt the calculations. This continuity makes the mathematics smooth and predictable, allowing for effective analysis and computation.