Math Help

Numerical Integration

In a Nut Shell: Not all functions yield to integration. For example, you may have a set
of experimental data for which you would like to perform an integration. In such cases,
one may use numerical integration.

Strategy: The general approach is to represent the “area” under the curve as a finite

collection of small areas. These areas may be in the shape of small rectangles, small

trapezoids, or perhaps small “parabolic bounded” areas under the “curve” of data.

Approximate area under y = f(x) using n small rectangles

b n

Area = ∫ f(x) dx = Σ f(x_i*) Δ x where

a i = 1

n is the number of subintervals

x_i* is any point in the ith subinterval (called the sampling point)

Three common sampling points for a rectangle are:

the left endpoint

the midpoint

the right endpoint

Click here for figures depicting the effect of sampling points on estimation of areas for

increasing and for decreasing functions.

Discussion and examples using rectangles to approximate the area under a curve

will be presented by the following three separate cases depending on the sampling

point selected for the rectangle.

Case 1: Procedure and an example using the right endpoint . Click here.

Case 2: Procedure and an example using the left endpoint. Click here.

Case 3: Procedure and an example using the midpoint. Click here.

For a better approximation one may break the area under the curve into n trapezoids

instead of n rectangles. This approximation leads to Case 4.

Case 4: Procedure and an example using trapezoids. Click here.

For an even better approximation one may break the area under the curve into n regions

bounded by parabolas instead of n trapezoids. This approximation leads to Case 5.

Case 5: Procedure and an example using parabolas. Click here.

Return to Notes for Calculus 1