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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

**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.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

More Lessons for Calculus

Math Worksheets

The area

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*]

** Example: **

Given that , evaluate

** Solution: **

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

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.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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