Clarification on a topic in Apostol's calculus text

[Srichan]

Hi all,
I am a newbie and this is my first post in PF. I have recently started on learning calculus and I'm using Apostol's calculus-I (second edition). I need some clarification on "The method of exhaustion for the area of a parabolic segment" (I 1.3 page 3). I'm able to follow the author till equation I.4 in page 6. After that there is a statement "For our purposes, we do not need the exact expressions given in the right-hand members of (I.3) and (I.4). All we need are the two inequalities" and there is equation - I.5 after that. Can someone explain me the reason how one can obtain equation I.5 (the inequality equation)?
Thankyou for reading my post

 

[Mark44]

Hi all,
I am a newbie and this is my first post in PF. I have recently started on learning calculus and I'm using Apostol's calculus-I (second edition). I need some clarification on "The method of exhaustion for the area of a parabolic segment" (I 1.3 page 3). I'm able to follow the author till equation I.4 in page 6. After that there is a statement "For our purposes, we do not need the exact expressions given in the right-hand members of (I.3) and (I.4). All we need are the two inequalities" and there is equation - I.5 after that. Can someone explain me the reason how one can obtain equation I.5 (the inequality equation)?
Thankyou for reading my post

I don't have Apostol's text, so it would be helpful if you told us what the inequalities are that are puzzling you.

 

[Srichan]

Thanks for the reply
http://books.google.co.in/books?id=vTpbq0UPDaQC&printsec=frontcover#v=onepage&q&f=false

above is the link for google book. The derivation for area of parabolic segment starts at page 3 and my doubt is equation I.5 (page 6) which is covered in preview.

I'm not sure how to present the derivation in the forum. I cannot scan or photograph the book as it will be against copyrights. If you find the above google book not appropriate, shall I make a word document reproducing the contents of the book ? please provide your comments

 

[Mark44]

Thanks for the reply
http://books.google.co.in/books?id=vTpbq0UPDaQC&printsec=frontcover#v=onepage&q&f=false

above is the link for google book. The derivation for area of parabolic segment starts at page 3 and my doubt is equation I.5 (page 6) which is covered in preview.

I'm not sure how to present the derivation in the forum. I cannot scan or photograph the book as it will be against copyrights. If you find the above google book not appropriate, shall I make a word document reproducing the contents of the book ? please provide your comments

"Fair use" would allow you to scan a paragraph or so without violating copyright protection. Best would be to type the inequalities directly into the input pane here. There are a number of symbols available to use when you click the $\Sigma$ button on the green row of icons above the input pane.

 

[mathwonk]

Apostol himself tells you how to prove inequality 1.5, namely either deduce it from the more precise equalities given prior to it, or use induction to prove it directly. have you tried induction? A remark: if you can't do this fairly easily, then I suspect this may not be the best book for you.

 

[Srichan]

Apostol himself tells you how to prove inequality 1.5, namely either deduce it from the more precise equalities given prior to it, or use induction to prove it directly. have you tried induction? A remark: if you can't do this fairly easily, then I suspect this may not be the best book for you.

I am able to prove the inequality by induction. I can also understand how the inequality can be derived from I.3 and I.4. I am able to follow till I.4 without any problem. But at I.5, I am not able to understand why the author wrote such a equation. Is that because he already takes A=b^3/3 wants to prove the same?

 

[Srichan]

Please refer the attachment for the problem

 

[mathwonk]

apostol is using equation 1.5 to prove that A = b^3/3. It seems to me this should be clear from reading his exposition. I.e. he gives an argument to deduce this and says "this is why Archimedes" knew that A = b^3/3. But perhaps you are asking how he thought of it, rather than how he proved it. I.e. perhaps you follow logically every step in Apostol's proof but not how he thought to make those steps. You ask us to tell you "how apostol obtained equation 1.5", but that could mean either how he proved it or how he he thiought of it. Indeed he is not telling you how he thought of these calculations. You might be ijnterested in Archimedes' work "the method" where he shows how he discovered using physics the facts that he afterwards proved logically.

 

[Srichan]

Thankyou mathwonk. I'll go through "the method"