Defining Definite Integrals

The definite integral is a fundamental concept in calculus that, on a basic level, represents the area under a curve.
A definite integral between two limits can be visualised as the area under the graph of a function, from the curve to the x-axis, between the two x-values.

Properties of Definite Integrals

The definite integral of a function between two bounds is negative if the function lies below the x-axis for that interval.
The integral from a to b is equal to negative the integral from b to a.
Linearity property: The integral of the sum of two functions is equal to the sum of their integrals over the same interval.
Constant multiple property: The integral of a constant multiplied by a function is that constant times the integral of the function.

In simple words, the definite integral from a to b is calculated by determining the indefinite integral at a and b and subtracting the former from the latter.
Definite integrals are computed using the Fundamental Theorem of Calculus which connects differentiation and integration.

Definite integrals are used in many fields of science and engineering to calculate quantities such as area, volume, and total amount of change.
Apart from calculating the area under curves, it is also used to compute the average value of a function on an interval.

Integration by Substitution and Integration by Parts are key techniques for carrying out more complex definite integrals.
If a function is continuous on the interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral from a to b of f(x) dx is F(b) - F(a). This is known as the evaluation theorem.
Definite integration can handle functions that are negative, and can handle functions that “go to infinity”, unlike indefinite integration.