Textbook ExpertVerified Tutor
13 Dec 2021
Given information
The function is
Step-by-step explanation
Step 1.
The midpoint rule (also known as the midpoint approximation) uses the midpoint of a subinterval:
where ![](data:image/png;base64,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)
We have that
, and
Therefore, ![](data:image/png;base64,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)
Divide the interval
into
subintervals of the length
with the following endpoints: