Understanding Definite Integrals

The definite integral is an integral which has definite upper and lower limits.
It’s often represented as ∫ab f(x) dx, where ‘a’ and ‘b’ are the lower and upper limits respectively.
The integral of a function between two limits measures the area under the curve between these points along the x-axis.
Unlike indefinite integrals, definite integrals do not include the arbitrary constant ‘C’ as the definite integral is taking into account a specific interval on the graph.
Definite integrals can be positive, negative, or zero, depending on the graph of the function.

Calculating Definite Integrals

To compute a definite integral, first calculate the indefinite integral or the antiderivative of the function f(x).
The resulting function, let’s say F(x), is then evaluated at the upper limit (b) and the lower limit (a), and the value at the lower limit is subtracted from the value at the upper limit.
This is often written as ∫ab f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).

Additive Property: The integral of the sum of two or more functions is equal to the sum of their integrals within the same limits.
Constant Multiple Property: The integral of a function multiplied by a constant is equal to the constant times the integral of the function.
Reversal of Limits: If the limits of the integral are reversed, then the sign of the integral switches. So, ∫ab f(x) dx = -∫ba f(x) dx.

Definite integrals can be used to find area, volume, central points, and many other useful things.
They are widely used in physics, engineering, and many areas of mathematics such as geometry, probability, and statistics.

Definite Integral of f(x): ∫ab f(x) dx = F(b) - F(a) where ‘F(x)’ is the antiderivative of ‘f(x)’.
Definite Integral of a Constant (c): ∫ab c dx = c(b - a).
Additive Property of Definite Integrals: ∫ab [f(x) + g(x)] dx = ∫ab f(x) dx + ∫ab g(x) dx.
Constant Multiple Property of Definite Integrals: ∫ab cf(x) dx = c ∫ab f(x) dx.