(a.)
To Show: The RRAM Riemann sum for the integral is
(a.)
Answer to Problem 70E
It has been shown that the RRAM Riemann sum for the given integral is
Explanation of Solution
Given:
The integral
Concept used:
The RRAM Riemann sum for the integral
Calculation:
The given integral is
Comparing
Partition
Put
Simplifying,
On further simplification,
Now,
Put
This shows that the RRAM Riemann sum for the given integral is
Conclusion:
It has been shown that the RRAM Riemann sum for the given integral is
(b.)
To Show: The sum obtained in part (a) can be written as
(b.)
Answer to Problem 70E
It has been shown that the sum obtained in part (a) can be written as
Explanation of Solution
Given:
The integral
Concept used:
Any term not containing the index variable can be taken out of the summation.
Calculation:
As determined previously, the sum obtained in part (a) is
Simplifying,
On further simplification,
Note that the index variable in the above summation is
This shows that the sum obtained in part (a) can be written as
Conclusion:
It has been shown that the sum obtained in part (a) can be written as
(c.)
To Show: The sum obtained in part (b) can be written as
(c.)
Answer to Problem 70E
It has been shown that the sum obtained in part (b) can be written as
Explanation of Solution
Given:
The integral
Concept used:
It can be shown by mathematical induction that
Calculation:
As determined previously, the sum obtained in part (b) is
Put
Simplifying,
On further simplification,
This shows that the sum obtained in part (b) can be written as
Conclusion:
It has been shown that the sum obtained in part (b) can be written as
(d.)
To Show:
(d.)
Answer to Problem 70E
It has been shown that
Explanation of Solution
Given:
The integral
Concept used:
Calculation:
As determined previously,
As determined previously,
Put
Simplifying,
On further simplification,
Continuing simplification,
Taking limit on both sides,
Simplifying,
On further simplification,
Put
Solving,
This is the required proof.
Conclusion:
It has been shown that
(e.)
To Explain: Why the equation in part (d) proves that
(e.)
Answer to Problem 70E
It has been explained why the equation in part (d) implies that
Explanation of Solution
Given:
The integral
Concept used:
If the interval
Calculation:
Partitioning
The
Choosing
It can be seen from the given integral
Then,
Put
Put
Put
As determined in part (d),
Put
This is the required proof.
Conclusion:
It has been explained why the equation in part (d) implies that
Chapter 5 Solutions
Advanced Placement Calculus Graphical Numerical Algebraic Sixth Edition High School Binding Copyright 2020
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