Understanding the Definite Integral | Calculation and Properties

a∫b f(x)dx

The expression “a∫b f(x)dx” represents the definite integral of a function f(x) over an interval [a, b]

The expression “a∫b f(x)dx” represents the definite integral of a function f(x) over an interval [a, b].

To understand this notation, let’s break it down:

– The symbol “∫” indicates integration. It represents the process of finding the area under the curve of a function between two points.
– “f(x)” represents the function being integrated. It can be any function in terms of the variable x.
– “dx” represents the differential of x. It signifies that we are integrating with respect to x.

Now, the values “a” and “b” provide the interval over which we are integrating. The lower limit, a, denotes the starting point of the interval, while the upper limit, b, represents the endpoint.

To calculate the definite integral, you need to follow these steps:
1. Find the antiderivative of the function f(x), denoted as F(x). The antiderivative is the reverse process of differentiation, and it is also called the indefinite integral.
2. Evaluate F(x) at the upper limit b, which gives you F(b).
3. Evaluate F(x) at the lower limit a, which gives you F(a).
4. Finally, subtract F(a) from F(b). This gives you the value of the definite integral of f(x) over the interval [a, b].

The result of the definite integral represents the accumulated area between the function f(x) and the x-axis over the interval [a, b]. If the result is positive, it indicates that the area above the x-axis is larger. But if the result is negative, it means that the area below the x-axis is greater.

The definite integral has a few important properties, such as the linearity property, where the integral of a sum of functions is equal to the sum of their integrals, as well as the fundamental theorem of calculus, which connects differentiation and integration by stating that the derivative of the definite integral of a function is equal to the original function.

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