# Calculus/Definite integral/Solutions

From Wikibooks, open books for an open world

1. Use left- and right-handed Riemann sums with 5 subdivisions to get lower and upper bounds on the area under the function from to .

**Lower bound:
Upper bound: **

2. Use left- and right-handed Riemann sums with 5 subdivisions to get lower and upper bounds on the area under the function from to .

**Lower bound:
Upper bound: **

3. Use the subtraction rule to find the area between the graphs of and between and

From the earlier examples we know that and that . From this we can deduce

4. Use the results of exercises 1 and 2 and the property of linearity with respect to endpoints to determine upper and lower bounds on .

In exercise 1 we found that

and in exercise 2 we found that

From this we can deduce that

5. Prove that if is a continuous even function then for any ,

From the property of linearity of the endpoints we have

Make the substitution . when and when . Then

where the last step has used the evenness of . Since is just a dummy variable, we can replace it with . Then