A Riemann sum is a method for approximating the area under a graph using rectangles and trapezoids.

A general Riemann sum with rectangles that approximates the area under the graph over the interval above can be described as follows

Where is a partition of and , and is some -value arbitrarily picked in the interval .

Types

Left Bounded (LRAM)

A left bounded Riemann sum is a Riemann sum where the value is the leftmost point of the partition. In left bounded Riemann sums typically the partitions are equal lengths.

A general left bounded Riemann sum with rectangles over the interval can be described as follows

Where is a partition of and .

Right Bounded (RRAM)

A right bounded Riemann sum is a Riemann sum where the value is the rightmost point of the partition. In right bounded Riemann sums typically the partitions are equal lengths.

A general right bounded Riemann sum with rectangles over the interval can be described as follows

Where is a partition of and .

Middle Bounded (MRAM)

A middle bounded Riemann sum is a Riemann sum where the value is the middlemost point of the partition. In middle bounded Riemann sums typically the partitions are equal lengths.

A general middle bounded Riemann sum with rectangles over the interval can be described as follows

Where is a partition of and .

Trapezoidal Approximation

The trapezoidal approximation is a Riemann sum that uses trapezoids instead of rectangles to approximate the area under a curve. In trapezoidal Riemann sums, the partitions are typically of equal length. The trapezoidal approximation can be viewed as the average of the LRAM and RRAM.

A general trapezoidal approximation with nn trapezoids over the interval [a,b][a, b] is given by:

where:

• is the width of each subinterval
• are the endpoints of the subintervals
• and are the values at the endpoints

Error

The error induced by trapezoidal Riemann sum can be bounded by the following equation

Where is a value such that for in , is a trapezoidal Riemann sum using points.