Improper Integral question: Convergence of 1/(x^p) from 0 to 1

[zachem62]

Homework Statement

Consider the function f(x)=1/(x^p).

When p>1, the integral of 1/(x^p) from 1 to infinity converges.


i) For what values of p does the integral of 1/(x^p) from 0 to 1 converge? (0<p<infinity, p does not equal 1).

ii) Confirm the answer by re-writing the integral of 1/(x^p) from 0 to 1 in terms of area and an integral in terms of y. Comment on any symmetry/asymmetry that this relation demonstrates).

Homework Equations

The Attempt at a Solution

i) The integral of 1/(x^p) from 0 to 1 is ((1^(-p+1))/(-p+1))-((0^(-p+1))/(-p+1))
When p>1, then 0 will have a negative power and the integral will diverge. Therefore the integral converges for 0<p<1.

ii) I have no clue how to do this part since I don't really understand what the question is asking when it says rewrite the integral in terms of area since the integral itself is about evaluating area. And I don't get the part where it asks to comment on symmetry.

PLEASE HELP!

 

[Oster]

I think they want you to integrate with respect to y instead of x.

 

[zachem62]

I think they want you to integrate with respect to y instead of x.

yeah it says rewrite the integral in terms of y and i had no problem getting that part. the part i didn't get is they ask me to rewrite the integral in terms of area...wtf does that even mean? when you take the definite integral that itself represents area doesn't it??

 

[Oster]

Hmm, I have no idea what 'in terms of area' means. Sorry. Ignore? =D