Definite Integral Definition
The definite integral is a fundamental concept in calculus that represents the accumulation of a function over a given interval
The definite integral is a fundamental concept in calculus that represents the accumulation of a function over a given interval. It allows us to calculate the area under a curve, the total value of a quantity, or the total change of a variable within a specific range.
The definite integral of a function f(x) over the interval [a, b] is denoted by ∫[a, b] f(x) dx, where dx represents the differential of x. The integral symbol (∫) represents the operation of integrating or summing up infinitesimal quantities.
To calculate the definite integral, we use the process of anti-differentiation or finding the antiderivative of the function. The antiderivative of a function f(x) is a function F(x) whose derivative equals f(x), i.e., F'(x) = f(x).
The definite integral is evaluated by subtracting the antiderivative at the lower limit (a) from the antiderivative at the upper limit (b):
∫[a, b] f(x) dx = F(b) – F(a)
In this context, F(x) is called the antiderivative (or indefinite integral) of f(x).
The definite integral has many applications in various fields, including physics, engineering, economics, and statistics. It can be used to calculate the area between curves, find the displacement of an object given its velocity, determine the total value of a function over a given range, or even solve problems involving probabilities.
It’s important to note that the definite integral is a numerical value and not a function, as it represents the cumulative quantity of a function over an interval, rather than the relationship between variables.
In summary, the definite integral is a mathematical operation that allows us to find the accumulation or total change of a function over a specific interval. It is denoted by ∫[a, b] f(x) dx and is evaluated by subtracting the antiderivative of the function at the lower limit from the antiderivative at the upper limit.
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