Integration Strips

Each strip contributes a rectangle whose area is the product of its width, Δx, and a function value selected within that subinterval. Adding these products creates a Riemann sum. The selected point therefore controls each rectangle’s height, while the partition controls how finely the curve is represented, allowing different approximations over the same interval.

As a partition is refined, the rectangles follow the curve over shorter subintervals rather than spanning larger portions of it. The resulting sum can approach the definite integral as Δx approaches zero. This limiting idea explains why rectangular estimates can represent an exact accumulated quantity, not merely a rough geometric approximation.

These approaches differ in where the function height is sampled within each subinterval: at the left endpoint, right endpoint, or midpoint. Because those choices produce different rectangle heights, they generally produce different sums. Comparing them provides a practical way to examine approximation behavior and study the effect of sample-point selection on error.

To construct an estimate, first specify the interval and divide it into subintervals. Determine each strip’s width, choose a sample point for each subinterval, evaluate the function there, and multiply the resulting height by the width. Summing all rectangle areas produces the numerical approximation that can be compared with the exact integral.

The interval, partition, strip width, sample-point rule, function values, and resulting products should be identified. Keeping these elements visible shows how the Riemann sum was formed and makes it possible to compare left, right, or midpoint estimates without changing the underlying function or interval.

A definite integral can represent an accumulated quantity, so strip sums can estimate distance, mass, or probability when the relevant function and interval are supplied. In each case, the rectangles provide a numerical route from local function values to a total quantity across the selected interval.

Calculate sums using different sample-point choices or a finer partition, then compare the resulting estimates. Differences reveal how sensitive the numerical result is to rectangle placement and strip width. This process connects computational approximation with the limiting behavior expected as Δx becomes smaller and supports systematic error analysis.