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.

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 - x^{2} from x = 0 to x = 2.

**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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