Calculus/Definite integral/Solutions

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