# Definite Integral - Limit of Riemann Sums

## The Problem of Areas

Riemann integrals are introduced as a concept using the example of finding the area of a circle from the areas of N-sided polygons inscribed in the circle. Signed area (positive above the x-axis, negative below) is introduced.

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## Partitions

Interval partitions are defined, including the concepts of mesh size and fine vs. coarse partitions.

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## Riemann Sums

Definition, including a discussion of partition choices when computing these sums.

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Definition, including the use of Riemann sums in finding the area under a curve.

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## The Riemann Integral

Definite integrals are defined. Includes an example using the function f(x) = x.

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## The Riemann Sum for the Exponential Function

Using Riemann sums to find the Riemann integral for the function f(x) = ex.

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## The Riemann Sum for xr

Using Riemann sums to find the Riemann integral for the function f(x) = xr.

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## Area and Notation

Definition of the definite integral as the area under a curve, including definition of the integrand.

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## Evaluating a Riemann Integral

Using upper sums to evaluate a definite integral.

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## Riemann Sums and Integrals

Three problems which involve evaluating Riemann sums and integrals.

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## Limit Definition of Integral

Evaluating an integral using the definition of an integral as the limit of sums.

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## Definite Integrals

Seven questions which involve using sigma notation for sums, computing Riemann sums for definite integrals, and evaluating limits by relating them to Riemann sums.

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