Q 10 - Calculus

Question

Consider the area between the graph of a positive function

f and the x-axis on an interval [a, b]. Explain why the

upper sum approximation for this area with n = 8 boxes

must be smaller than or equal to the upper sum approximation

with n = 4 boxes. It may help to sketch some

examples.

Step-by-Step Solution

Answer

To explain, the upper sum approximation with $n=8$ boxes for this region must be smaller or equal to the upper sum approximation with $n=4$ boxes.

Step 1: Solution Explanation

On the interval $[a,b]$, consider the area between the graph of a positive function $f$ and the $x-$axis.

Step 2:

In two separate scenarios, the number of rectangles taken is $n=8$.

The top total is always bigger than or equal to the signed area in question.

When, the number of sub intervals that is the number of rectangles increases, the width of the subintervals reduces which contributes to a lower sum.

As the number of sub intervals, or rectangles, reduces, the width of the sub intervals grows, resulting in a larger sum.

As a result, the area upper approximation with $n=8$, $n=4$ must be smaller or equal to the upper sum approximation with $n=8$.