Solving Double Integrals: √(1=y^2) & √(2x=x^2)

[Audax Dreik]

This might be somewhat of a mundane question but I can't seem to figure it out. It has to do with the limits of integration for a double integral. The initial integral is as follows...

1 √(1=y^2)
∫ ∫ 1/(1+x^2+y^2) dx dy
0 0

I hope the formatting on that doesn't get screwed up. Anyway, the point of the exercise is to convert this to polar and do the integral then. I can convert the equation easy enough, especially due to the x^2 and y^2 just turning into an r^2, however my question is what in the world is that one limit supposed to be? dx is first so it's like saying x = √(1=y^2)? I would imagine it is something that will convert to polar nicely since these are specially engineered excercises but I'm just not sure what to do with it with that = sign in there. There's also a second question with a similar limit y = √(2x=x^2). Sorry if this is a stupid question but I haven't encountered this notation before and it puzzles me.

 

[Spectre5]

the best way is to draw that region to convert on the x-y plane...

then look at it and determine the limits for dr and dtheta

 

[Kurdt]

just think of how the polar coordinates are connected to the normal rectangular coordinates.

x=r*cos(theta)
y=etc

 

[Kurdt]

actually come to think if that's the equation of a circle r will remain constant and theta will vary by 2pi. ok I'm out of practise maybe i should be quiet.

 

[Spectre5]

It is the part of the circle in the first quadrant only :)
the region that is

 

[Audax Dreik]

Sorry, maybe I wasn't clear enough. I don't understand what the limits mean. On the y-axis it's going from 0 to 1, I can see that but what about the x axis? What is meant by integrating from 0 to √(1=y^2)? I can't even figure out how to visualize this in rectangular coordinates because that "=" sign in there is confusing mean. I just don't understand the notation.

 

[Spectre5]

=...I am sure that is supposed to be a - lol

EDIT: a minus, not a plus..

 

[Kurdt]

i thought that was supposed to be a minus sign if so it comes from the equation of a circle:

x**2+y**2=1

can be rearranged into the upper limit of the x-axis quite easily

 

[Audax Dreik]

...
-_-;; Sorry for posting such a stupid question, it's just that he made that same typo twice and I thought it was some kind of notation... Heheh... *quietly walks away

 

[Spectre5]

oh, sorry about that...yea, I meant a minus sign :)