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

The upper approximation for the area with $n=8, n=4$ be smaller than or equal to the upper sum approximation with $n=8$

Step 1: Given Information

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.

Step 2: Explanation

Consider the area between the graph of a positive function $f$ and the $x$axis on an interval $[a, b].$

The number of rectangles taken in two different cases are $n=8$

The upper sum is always greater than or equal to the actual signed area.

When, the number of sub intervals that is the number of rectangles increases, the width of the subintervals decreases which causes to a smaller sum.

The number of sub intervals that is the number of rectangles decreases, the width of the sub intervals increases which causes to a greater sum.

Therefore, the upper approximation for the area with$n=8, n=4$ be smaller than or equal to the upper sum approximation with $n=8$.