How do you find the area under a diagonal hyperbola xy=a where a is a constant?

^topic

 

Just, what is the general process for finding the area? Say for a=3.

 

Perhaps I should have specified. I knew how to find the are under the curve from a to b, where a is not equal to zero, but wanted to find the area of the curve in either the 1st or 3rd quadrant (that is, from 0 to b or from -b to 0).

 

Nope, sorry about that, I should have used different variables. The second a (as in from a to b) just represents that I wasn't asking about the integral from c to b where c≠0 and was concerned with the integral from 0 to b.

 

Right, as I said I got this result when I tried myself. But I wanted to know what to do when c=0.

 

That's the problem, and it's what made me think I needed a new approach. Because obviously a hyperbola in the 1st quadrant doesn't have infinite area as it has asymptotes at the x and y axis. Is there perhaps another way to figure this out?

 

Because it has a finite area except at the values x=0 and y=0 which approach a negligible area along the axes. If you graph it, the area is significantly less than a squared. Are there at least any approximation methods (maybe perturbation) out there that would work?

 

Wait, but for the second example, isn't the following true:
[itex]\int[/itex][itex]^{\infty}_{1}[/itex][itex]\frac{1}{x}[/itex]dx[itex]\equiv[/itex]-ln(1)+[itex]lim_{x->{\infty}}[/itex]ln(x)=0

?


Also, are there any clever approximations for this area? Would translating the figure into say (x-3)(y-3)=a make any difference (no more singularity)? How would I find the area under a different type of hyperbola (is there a method that works for all hyperbolas that aren't in the form xy=1)? Thanks.

 

Yeah, your right, but Wolfram was giving me a different result a minute ago, or maybe I made a typing mistake. Anyway, can you help with my other questions (sorry for so many)? There has to be some way to approximate it, especially considering that the hyperbola all but approaches the axes at an upper limit of 10*a.

 

These questions, you may not have noticed them because they were on the first page. Thank you for clearing up the first matter, I did not mean to imply that anyone other than myself was making a mistake.