Actual integration of a function

[courtrigrad]

Hello all

I am having trouble integrating the function x^a. Take in consideration that we are not using any rules yet, but actually taking the passage of the limit. My question is:

int (from a to b) x^a dx why would it be inconvenient to divide the interval into equal parts? In the book it says we divide the interval as follows:

a, aq, aq^2, ..., aq^n-1, aq^n = b. (which is the geometric progression)

The answer is: (1/(a+1)(b^a+1 - a^a+1). But in the integration of x^2, we divide the interval from a to b in equal lengths of b/n. Why is this? Finally, do you think it is worth the time to do every single problem say in Courant's calculus book?

Any help would be greatly appreciated!

Thanks

 

[NateTG]

I am having trouble integrating the function x^a. Take in consideration that we are not using any rules yet, but actually taking the passage of the limit. My question is:

int (from a to b) x^a dx why would it be inconvenient to divide the interval into equal parts? In the book it says we divide the interval as follows:

a, aq, aq^2, ..., aq^n-1, aq^n = b. (which is the geometric progression)

The answer is: (1/(a+1)(b^a+1 - a^a+1). But in the integration of x^2, we divide the interval from a to b in equal lengths of b/n. Why is this?

You can divide the interval in any way you like. The idea is typically to select intervals so that the difference between lower and upper bounds is easy to calculate.

Finally, do you think it is worth the time to do every single problem say in Courant's calculus book?

I'm not familiar with the particular book, but for the more 'serious' calculus texts like Apostol it is at least worth looking at all of the questions to see if you can do them.

 

[matt grime]

I think you're best off rewriting so that a has one meaning alone, and that the other terms are consistent (eg, n).

One of the bits of integration theory is that we may choose the division to suit our purposes. If we're integrating x^n, where n is a whole number, then equal sudivisions into r equal parts will lead to a sum n'th powers of integers, which we may well know how to sum, particularly if n=2.

However, it is easier to sum geometric progressions, which is why the book chooses them, I suppose.

 

[arildno]

NateTG: From what I've heard, Courant's book is at least as serious as Apostol's.

 

[NateTG]

NateTG: From what I've heard, Courant's book is at least as serious as Apostol's.

It may not have been sufficiently clear, but I meant to make a general rather than specific comparison when I wrote more serious. I.e. I meant to say that if the calculus text is a serious calculus text, then it's a good idea to (at least) look at all of the problems.

 

[ReyChiquito]

IMO Courant's Book is even better than Apostol's, in fact, i think is the most complete calculus book around.