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

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The discussion centers on understanding the limits of integration for a double integral involving the equations √(1=y^2) and √(2x=x^2). The user is confused by the notation, particularly the equal sign, which they later clarify should be a minus sign, indicating a circular equation. They express difficulty in visualizing the integration limits in rectangular coordinates and seek guidance on converting to polar coordinates. The key takeaway is that the region of integration corresponds to a quarter circle in the first quadrant, with the correct limits derived from the equation x² + y² = 1. The conversation highlights the importance of accurately interpreting mathematical notation in solving integrals.
Audax Dreik
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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 excercise 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.
 
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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
 
just think of how the polar coordinates are connected to the normal rectangular coordinates.

x=r*cos(theta)
y=etc
 
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.
 
It is the part of the circle in the first quadrant only :)
the region that is
 
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.
 
=...I am sure that is supposed to be a - lol

EDIT: a minus, not a plus..
 
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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
 
...
-_-;; 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
 
  • #10
oh, sorry about that...yea, I meant a minus sign :)
 

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