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 [a, b]

The following diagram gives some properties of the definite integral. Scroll down the page for more examples and solutions.

 

Example:

Given that , evaluate

Solution:

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.

Approximating a Definite Integral Using Rectangles
This video shows how to use 4 rectangles and left endpoints as well as midpoints to approximate the area underneath 16 - x² from x = 0 to x = 2.

• Show Step-by-step Solutions

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.

• Show Step-by-step Solutions

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