2.
Then by our rulesfor definiteintegrals.
- Then. Hence
- Since is continuouson and then we can select suchthat
is the minimumvalueof and is the maximumvalueof in Thenwe can consider
as a lowersum and as an uppersum of from to Hence
5.
- By substitution,we have:
- By division,we have
- When is closeto then both and are closeto by the continuityof
- Hence Similarly, if then Hence,
- By the definitionof the derivative,we havethat
for every Thus, is an antiderivativeof on
ReviewQuestions
In problems#1–4,sketchthe graphof the function in the interval Thenuse the Fundamental
Theoremof Calculusto find the area of the regionboundedby the graphand the -axis.