Question

How is integration related to derivation?

Short answer

Answer: Integration and derivation are related through the Fundamental Theorem of Calculus, which connects these two concepts by presenting integration as the reverse process of differentiation. The derivative of a function represents its rate of change, while the integral represents the accumulated value of the function over a specified interval. Antiderivatives play a crucial role in demonstrating this relationship, as they are functions that, when derived, produce the original function. Essentially, differentiation and integration are inverse operations, and the connection between them can be seen through the process of finding the antiderivative of a function.

Step-by-step solution

Understand Derivation

Derivation is the process of finding the derivative of a function, which represents the rate of change of the function at any given point. The derivative, denoted as f'(x) or \frac{d}{dx}f(x) , is a new function that gives the instantaneous rate of change of the original function f(x) at each point x.

Understand Integration

Integration is the process of finding the integral of a function, which represents the accumulated value of the function over a specified interval. It is the reverse process of differentiation. There are two types of integration: definite and indefinite integration. Definite integration computes the signed area under the function's curve between two points, while indefinite integration finds the antiderivative (inverse function to the derivative) that represents a family of functions differing only by a constant.

Explain Antiderivatives

Antiderivatives are functions that, when derived, produce the original function. In other words, if F(x) is the antiderivative of f(x), then F'(x) = f(x). This means that integration reverses the process of differentiation. For example, if the derivative of F(x) is f(x), then F(x) is an antiderivative of f(x). The indefinite integral of f(x)dx, denoted by \int{f(x)dx}, is the set of all antiderivatives of the function f(x).

Use the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects integration with derivation, stating that differentiation (finding the derivative) and integration (finding the antiderivative) are inverse operations. Following the theorem: if a function f(x) is continuous over the interval [a, b] and F(x) is an antiderivative of f(x) over the interval [a, b], then the definite integral of f(x) from a to b is given by F(b) - F(a) or \int_{a}^{b}{f(x)dx} = F(b) - F(a).

Show a Function example

Let's use a simple function, f(x) = x^2, as an example. The derivative of f(x) is f'(x) = 2x. Now, let's find the antiderivative of f'(x). The antiderivative, F(x), would be the integration of f'(x), which is \int{2x dx} = x^2 + C, where C is the constant of integration. Notice how F(x) represents the original function, f(x), plus the constant C, thus showing the relationship between integration and derivation.