Q 9 - Calculus

Question

Explain why the upper sum approximation for the area

between the graph of a function f and the x-axis on [a, b]

must always be larger than or equal to any other type of

Riemann sum approximation with the same number n of

rectangles.

Step-by-Step Solution

Answer

To explain, any sort of Riemann sum approximation with the same number $n$ of rectangles must always be larger than or equal to the upper sum approximation for the area between the graph of a function $f$ and the $x-$axis on.

Step 1: Introduction

Consider that the area between a function's graph and the x axis on [a,b] must always be larger than or equal to any other form of Riemann sum approximation with n number of rectangles.

Step 2: Explanation

The maximum height of the rectangle is taken into account while calculating the upper sum.

As a result, the heights utilized in the upper sum are always greater than or equal to the heights considered in any other Riemann sum types.

As a result, the upper sum approximation for an area between a function's graph and the $x-$axis on $[a,b]$ must always be greater than or equal to any other form of Riemann sum of approximation with n rectangles.