Textbook ExpertVerified Tutor
6 Jan 2022
Given information
Given that,
and 
for 
Step-by-step explanation
Step 1.
If 
is the number of rectangles, then the width of a rectangle is
is the number of rectangles, then the width of a rectangle is
We wanted to use left endpoints then the first rectangle would be height 
because they say , and then it would be the second rectangle ( 2nd term) that could be made arbitrarily large. But what we need to do is just for the first rectangle, we pick a sample point closer to , such as 
. It does not matter what the other rectangles use as we only need to look at the first term since that is the one that gets crazy.
because they say , and then it would be the second rectangle ( 2nd term) that could be made arbitrarily large. But what we need to do is just for the first rectangle, we pick a sample point closer to , such as . It does not matter what the other rectangles use as we only need to look at the first term since that is the one that gets crazy.Now, the area of the first rectangle would then be
is positive on the interval, so the whole sum is greater than the area of the first rectangle while is a finite number.
Again, if we increase , towards infinity then
, towards infinity then
The first term, and the entire sum go to infinity. A function is not integrable on a certain interval if the sum would go to 
(or 
).
(or ).
Single Variable Calculus: Early Transcendentals
4th Edition, 2018
Stewart
ISBN: 9781337687805
Solutions
Chapter
Section
- Problem 1
- Problem 2
- Problem 3
- Problem 4a
- Problem 4b
- Problem 5a
- Problem 5b
- Problem 5c
- Problem 6a
- Problem 6c
- Problem 7
- Problem 8
- Problem 9
- Problem 10
- Problem 11
- Problem 12
- Problem 13
- Problem 14
- Problem 15
- Problem 16
- Problem 17
- Problem 18
- Problem 19
- Problem 20
- Problem 21
- Problem 22
- Problem 23
- Problem 24
- Problem 25
- Problem 26a
- Problem 26b
- Problem 26c
- Problem 26d
- Problem 27
- Problem 28
- Problem 29
- Problem 30
- Problem 31a
- Problem 31b
- Problem 31c
- Problem 31d
- Problem 32a
- Problem 32b
- Problem 32c
- Problem 33
- Problem 34
- Problem 35
- Problem 36
- Problem 37
- Problem 38
- Problem 39
- Problem 40
- Problem 41
- Problem 42
- Problem 43
- Problem 44
- Problem 45
- Problem 46
- Problem 47a
- Problem 47b
- Problem 47c
- Problem 47d
- Problem 47e
- Problem 48a
- Problem 48b
- Problem 48c
- Problem 48d
- Problem 48e
- Problem 49
- Problem 50
- Problem 51
- Problem 52
- Problem 53
- Problem 54
- Problem 55
- Problem 56
- Problem 82a
- Problem 82b
- Problem 83
- Problem 84