CK12 Calculus - Single Variable

2.

Then by our rulesfor definiteintegrals.

3. Then. Hence


4. 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.

6. By substitution,we have:


7. By division,we have


8. When is closeto then both and are closeto by the continuityof


9. Hence Similarly, if then Hence,


10. 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.