Rectangular Approximation Method and the Trapezoidal Rule

Sums of rectangular areas are called Riemann Sums. In practice we choose one of three methods: Left-endpoint Rectangular Approximation Method, Right-endpoint Rectangular Approximation Method, or Midpoint Rectangular Approximation Method. Another method is called the Trapezoidal Rule. The midpoint method and the trapezoidal rule are brought out in the solutions to the Rectangular Approximation Method and the Trapezoidal Rule.

A few things to remember with these methods:

1. Widths won't always be one and they won't necessarily be the same for each sub-interval.

2. LRAM under approximates when the function is increasing and over approximates when the function is decreasing.

3. RRAM is the opposite. It over approximates when the function is increasing and under approximates when it is decreasing.

4. Trapezoids over approximate when the function is concave up, but under approximate when it is concave down. When a function is linear, trapezoids are an exact fit.


This example demonstrates LRAM and RRAM in a very simple context.