Calculate the areas under the following curves using the method of exhaustion:
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The following graphs show the lower and upper sums, respectively:So, now we need to compute the lower and upper sums. For the lower sums we add up the area of all of the blue rectangle. Each rectangle has width (since we have divided the interval into equal pieces) and its height is the value of the curve on the left side of the rectangle (since the curve is drawn so that the left corner rests on the curve). Therefore, we have
For the upper sums the width of each rectangle is still , but the height is now given by the function value on the right corner. So we have,
Now, we are given the identity
and so, following the example in Apostol,
Now, we multiply each term in the inequality by so the sums on the far left and far right become our and above, and we can simplify the term in the middle. Thus, for all we have
Now, we must show that if for all then . We accomplish this by showing that and both lead to contradictions.
Suppose . Then, since we can compute,
However, this cannot hold for all since the term on the right is a constant (depending on ), so we can always choose some that is larger since the positive integers are unbounded above.
Next, suppose . Then,
But, again, this cannot be true for all positive integers since the positive integers are unbounded above. Therefore, both and lead to contradictions. Hence, we must have
 .
The following graphs show the lower and upper sums, respectively:As in part (a), we need to compute the lower and upper sums. For the lower sums we add up the area of all of the blue rectangle. Each rectangle has width (since we have divided the interval into equal pieces) and its height is the value of the curve on the left side of the rectangle (since the curve is drawn so that the left corner rests on the curve). Therefore, we have
For the upper sums the width of each rectangle is still , but the height is now given by the function value on the right corner. So we have,
Now, we are given the identity
and so, following the example in Apostol,
Now, we multiply each term in the inequality by so the sums on the far left and far right become our and above, and we can simplify the term in the middle. Thus, for all we have
Now, we must show that if for all then . We accomplish this by showing that and both lead to contradictions.
Suppose . Then, since we can compute,
However, this cannot hold for all since the term on the right is a constant (depending on ), so we can always choose some that is larger since the positive integers are unbounded above.
Next, suppose . Then,
But, again, this cannot be true for all positive integers since the positive integers are unbounded above. Therefore, both and lead to contradictions. Hence, we must have

The following graphs show the lower and upper sums, respectively:This will be very similar to parts (a) and (b) except we have the factor of instead of 2 or 3. First, we need to compute the lower and upper sums. For the lower sums we add up the area of all of the blue rectangle. Each rectangle has width (since we have divided the interval into equal pieces) and its height is the value of the curve on the left side of the rectangle (since the curve is drawn so that the left corner rests on the curve). Therefore, we have
For the upper sums the width of each rectangle is still , but the height is now given by the function value on the right corner. So we have,
Now, we are given the identity
and so, following the example in Apostol,
Now, we multiply each term in the inequality by so the sums on the far left and far right become our and above, and we can simplify the term in the middle. Thus, for all we have
Now, we must show that if for all then . We accomplish this by showing that and both lead to contradictions.
Suppose . Then, since we can compute,
However, this cannot hold for all since the term on the right is a constant (depending on ), so we can always choose some that is larger since the positive integers are unbounded above.
Next, suppose . Then,
But, again, this cannot be true for all positive integers since the positive integers are unbounded above. Therefore, both and lead to contradictions. Hence, we must have
 .
The following graphs show the lower and upper sums, respectively:Again, we need to compute the lower and upper sums. For the lower sums we add up the area of all of the blue rectangle. Each rectangle has width (since we have divided the interval into equal pieces) and its height is the value of the curve on the left side of the rectangle (since the curve is drawn so that the left corner rests on the curve). Therefore, we have
For the upper sums the width of each rectangle is still , but the height is now given by the function value on the right corner. So we have,
Now, we are given the identity
and so, following the example in Apostol,
Next, to get the inequalities we want (with ) it takes a bit more work than in parts (a) – (c) since we now have the term to deal with. First, we multiply each term in the inequality by to obtain
Now, we must show that if for all then . We accomplish this by showing that and both lead to contradictions.
Suppose . Then, since we can compute,
However, this cannot hold for all since the term on the right is a constant (depending on ), so we can always choose some that is larger since the positive integers are unbounded above.
Next, suppose . Then,
But, again, this cannot be true for all positive integers since the positive integers are unbounded above. Therefore, both and lead to contradictions. Hence, we must have
 .
Finally, we follow the same methods as in parts (a) – (d) to compute the lower and upper sums,
Then, similar to part (d), we have
Therefore,
Now, we must show that if for all then . We accomplish this by showing that and both lead to contradictions.
Suppose . Then, since we can compute,
However, this cannot hold for all since the term on the right is a constant (depending on ), so we can always choose some that is larger since the positive integers are unbounded above.
Next, suppose . Then,
But, again, this cannot be true for all positive integers since the positive integers are unbounded above. Therefore, both and lead to contradictions. Hence, we must have
In e. you have made an error, in my opinion:
s_n = a(\frac{b}{n})^{3}(0^{2} + 1^{2} + (n1)^{2}) + (\frac{b}{n})nc
See, the rectangles under the curve begin with k=0, not k=1. thus, \frac{b}{n} is added up *n times* not *n1 times*
You did the same mistake in question d. but you seem have added it up *n times* anyway, which would have been *n1 times* instead if you followed your reasoning there logically.
Maybe I am underestimating sadistic math teachers, but it seems like once you have found an expression for the area of one of the rectangular segments given the new ordinate, you can realize that the problem is the same plus some constant multiplicative (and then additive) factor. Would treating it like this still be seen as making too much of an assumption for the type of course that would be built around this text?
I think that it is necessary to make a precission in (e) with respect to the sign of “a”. Given b>0, it is necessary to assume a>0, because otherwise the inequalities obtained are not correct. That is, if a<0, when we multipliy by "a", the inequalities go in the other way.
This precission is also necessary in all the other exercises where the function hast the form ax^m+c.
In 1.d and 1.e both when you are contradicting A > (2b^3)/3 + b , when substituing Sn – sn you replace with 2b^3/n + b ( wouldn’t this be b/n and not b ) and if not can you explain why i am a little confused on this part.
I am confused by the answers for 1d and 1e, because I do not understand why (2x^2 + 1) and
(ax^2 + c) are not treated as single units, so that the area of the rectangle is, for example, in the first instance, x(2x^2 +1) = 2x^3 + x. Then, using the formula A_(parabola) = b^3/3, we would get (2x^3 + x)/3.
Sorry to be so obtuse, but I just don’t understand the WHY.
There is a error in 1.c. Then number 3 should be 12.
Hi,
In the exercise 1.d and 1.e when you get to the inequalities (that end up with :
sn< (2(b`3)/3)+b)<Sn) first, you multiply each term in the inequality by: (b/n. 2.(b/n)`2) and then you add (nb/n). So far it's ok , but in the third step on the left side we have ((n1).b/n) instead of (nb/n) that we add in the previous step. what is the arithmetic done there that allows you to write that?. Of course we need that to get the area of the inner sums, but in terms of arithmetic i don't know how you back that up without affecting the left side of the inequality. Im not sure how can you get to that result without affecting the right side of the inequality. I tried to treat them separately and i have two different area assumptions that don't allow me to put them in the same inequality because when i try to build the inner area sums sn i have to add ( b/n ) to achieve
((n1).b/n).
I hope it's clear enough,
Thank you
Hi, I also want to type up my own solution but I had some troubles with the graph. Are you using the pfgplots for these graphs? I am new to latex and cannot figure out how to graph.. Any helps will be appreciated.
I think I found a mistake, in d) You state Snsn=(2b^3/n)+(b/n) but I get Snsn=(2b^3/n). Both yield the same overall result showing there is a contradiction with the unbounded n. Am I wrong or did I spot something? Because if I did I will feel pretty cool.
Hi Rori! Thanks for the great work! I have a question regarding Part (d). In the inequalities that you have written following the statement: “Next, to get the inequalities we want (with s_n < \cdots < S_n) it takes a bit more work than in parts (a) – (c) since we now have the +1 term to deal with. First, we multiply each term in the inequality", there seems to be a slight inconsistency. Inequality 2 arises as you have added nb/n throughout. However, in Inequality 3, for the leftmost identity, you change nb/n to (n1)b/n. Now, the leftmost identity should have (n1)b/n since there are n1 terms for the leftmost identity. Nevertheless, the result that you have in Inequality 3 is not implied from Inequality 2 unless you can show nb/n = 0, such that you can drop that term and you end up (n1)b/n as the last term in your leftmost identity. But this is not the case.
I hope you understood my issue! Specifically, I am wondering, for the leftmost identity, how did you end up with Inequality 3 from Inequality 2. How or why did you drop the nb/n term?
Thank you so much!
Hi Rori,
I understood my confusion. Your solution is correct. Sorry for the inconvenience!
I think you were right but the problem is the definition of Sn and sn.
For Sn you must take the “right side of the rectangle” (the height is given by the end of the interval for f(x)) so your addition goes from (b/n) to b, but for sn you must take the “left side” (the height is given by the begining of the interval for f(x)) hence the addition goes from 0 to bb/n. Both additions have n terms (just different heights) but if c=0 the first rectangle of sn cancels out (height=0), else you get height=c and it adds a term to the sum. Therefore the last term of Sn and sn should be nb/n instead of (n1)b/n for the latter.
Hi Subhankar, can you please tell me how you cleared up your confusion? I don’t see how one inequality follows from the next, i.e. how we can go from nb/n to (n1)b/n without subtracting b/n from the other side
it is inequality (<) , so (n1)b < nb, that is all