Understanding the Definite Integral: A Fundamental Concept in Calculus and How to Compute it

Theorem: the definite integral of a continuous function on [a,b]

The definite integral of a continuous function on the interval [a, b] is a fundamental concept in calculus

The definite integral of a continuous function on the interval [a, b] is a fundamental concept in calculus. It is denoted by ∫[a,b] f(x) dx, where f(x) represents the integrand, and dx represents the differential of x.

To understand the definite integral, we first need to understand the concept of an antiderivative. An antiderivative of a function f(x) is a function F(x) whose derivative is equal to f(x). This is denoted by F'(x) = f(x).

Now, let’s consider a continuous function f(x) defined on the interval [a, b]. The definite integral ∫[a,b] f(x) dx represents the signed area between the graph of f(x) and the x-axis over the interval [a, b]. The signed area is positive if the graph is above the x-axis and negative if it is below.

To compute the definite integral, we can use the fundamental theorem of calculus. According to the first part of the fundamental theorem of calculus, if F(x) is any antiderivative of f(x) on [a, b], then the definite integral is given by:

∫[a,b] f(x) dx = F(b) – F(a)

In other words, to evaluate the definite integral of a continuous function, we find an antiderivative, F(x), of the integrand and then evaluate it at the upper limit, b, and subtract the value of F(x) at the lower limit, a.

It is important to note that the value of the definite integral does not depend on the choice of the antiderivative. Any antiderivative will give the same result.

To compute the definite integral, we can follow these steps:
1. Find an antiderivative of the function f(x), denoted by F(x).
2. Evaluate F(x) at the upper limit, b, to get F(b).
3. Evaluate F(x) at the lower limit, a, to get F(a).
4. Subtract F(a) from F(b) to obtain the value of the definite integral.

It is also worth mentioning that if the integrand has any points of discontinuity within the interval [a, b], the definite integral may not exist. In such cases, the integral is said to be improper, and a different approach is required to evaluate it.

In summary, the definite integral of a continuous function on the interval [a, b] represents the signed area between the graph of the function and the x-axis. It can be computed using the fundamental theorem of calculus by finding an antiderivative of the function, evaluating it at the upper and lower limits, and subtracting the values.

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