Definite Integral

The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of the approximating rectangles.

Definition of a Definite Integral

Let f be a function that is continuous on the closed interval [a, b]. The definite integral of f from a and b is defined to be the limit

where

is a Riemann Sum of f is [a, b]

Properties of Definite Integral

We assume that f and g are continuous functions.

Example:

Given that , evaluate

Solution:

Videos

Approximating Area Using Rectangles
When finding the area under a curve for a region, it is often easiest to approximate area using a summation series. This approximation is a summation of areas of rectangles. The rectangles can be either left-handed or right-handed and, depending on the concavity, will either overestimate or underestimate the true area.

The Definite Integral
The definite integral is an important operation in Calculus, which can be used to find the exact area under a curve. The definite integral takes the estimating of approximate areas of rectangles to its limit by using smaller and smaller rectangles, down to an infinitely small size.

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