Question

Explain why it makes sense that every Riemann sum for a continuous function fon an interval$[a,b]$ approaches the same number as the number n of rectangles approaches infinity. Illustrate your argument with graphs.

Short answer

When a result, as the number n of rectangles approaches infinity, any Reimann sum for a continuous function f on an interval $[a,b]$approaches the same value.

Step-by-step solution

Step 1. Given information

They had given that all the Riemann sums for a function approachessame number as the napproaches to infinity.

Step 2. Prove

A Reimann sum is a discrete sum of rectangle areas with each rectangle being infinitely thin.

The finite sum of things becomes the $(\int )$integral as n approaches infinity, accumulating everything from $x=a$to$x=b$.

When a result, as the number n of rectangles approaches infinity, any Reimann sum for a continuous function f on an interval $[a,b]$approaches the same value.