Finding Upper and Lower Sums for a Region In Exercises 41-44, find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals
To calculate: The upper and lower sums for the region bounded by the graph of the provided function in the interval and -axis.
Given: The provided function is in the interval .
And the number of subintervals to be used in the calculation of area is .
Formula used: The sum of first natural is given by the formula:
The sum of a constant times is written as:
Lower sum is calculated by formula: , Where are the left endpoints.
And the upper sum is calculated by: , where are the right endpoints.
Calculation: To begin, partition the interval into subintervals, each of width
Left endpoints ,
The lower sum can be calculated by the formula:
Where, are the left end points.
By the left endpoints, the lower sum is:
Put value of ,
Factor out from the sum,
Split the summation into parts to use summation formulas:
Apply the formulas:
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