Method of exhaustion for the area of a parabolic segment (Apostol)

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Homework Statement

Hi all, I'm starting from page one of Apostol's Calculus Vol. 1 (ed. 2) to try and refresh and advance my maths while I'm on exchange and can't take maths courses. I'm an economics undergrad but I'm planning on switching to Maths/Computer Science if I can show myself I've got it in me. If I study this book like I intend to over the next 6 months I'm going to have a whole lot of questions, so I'd like to say in advance how grateful I am that a resource like this exists and that there are people who spend their time answering these questions.

From the exercises 1.4 1.e)

This is a very simple question but it's difficult to write all the necessary bits out on the computer. I have a feeling I probably don't really need to but just in case...

We are shown that for the function f(x)=x[tex]^{2}[/tex] the area of the rectangles used to find the area is given by (b/n)(kb/n)² = (b³/n³)k², where b/n is the constant length of the base and (kb/n)² is the length of the rectangle (ie. the y-value) at the point kb/n.

We are also shown that 1² + 2² + ... + (n-1)² < n³/3 < 1² + 2² + ... + (n)² and that multiplying all sides by (b³/n³)k² gives us s_n < b³/3 < S_n, telling us that therefore the area = B³/3

Question 1. e) from Exercises 1.4 asks to find use the same method to find the Area for the function ax² + c

The Attempt at a Solution

I said that the area of the rectangles would have the equation (b/n)(a(kb/n)²+c) = ak²b³/n³ + bc/n, from here I'm completely stuck as to how to get the k² out of the equation so that I can use the identity. The answet given in the book is Area = ab³/3 + bc and I have tried to work backwards from here but without success. Hopefully what I've written here makes sense (took bloody ages).

p.s. completely off-topic, I found a local beer with an especially apt name tonight at the supermarket: http://www.flickr.com/photos/62689184@N05/5709251282/

 

[mighty2000]

It's great to hear that you are taking the initiative to refresh and advance your math skills while on exchange. I can attest to the importance of continuously learning and challenging oneself in order to reach new heights in your field.

In regards to your question about finding the area for the function ax^2 + c, you are on the right track with your attempt at a solution. To get the k^2 out of the equation, you can use the fact that (kb/n)^2 = (k^2b^2)/n^2. This will allow you to simplify the expression and use the identity given in the book to find the area.

I hope this helps and keep up the great work! Also, that beer looks very interesting. Enjoy it responsibly while you continue your studies. Best of luck to you on your academic journey.