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

  • Thread starter Audax Dreik
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In summary, the excercise is to convert the equation of a circle into polar coordinates and determine the limits for dr and dthetajust think of how the polar coordinates are connected to the normal rectangular coordinates. The limit for the y-axis is found by rearranging the equation of a circle into upper limit of the x-axis quite easily.
  • #1
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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  • #2
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
 
  • #3
just think of how the polar coordinates are connected to the normal rectangular coordinates.

x=r*cos(theta)
y=etc
 
  • #4
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.
 
  • #5
It is the part of the circle in the first quadrant only :)
the region that is
 
  • #6
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.
 
  • #7
=...I am sure that is supposed to be a - lol

EDIT: a minus, not a plus..
 
Last edited:
  • #8
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
 
  • #9
...
-_-;; 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 :)
 

1. What is a double integral?

A double integral is a mathematical concept used to calculate the volume under a surface in two dimensions. It involves integrating a function over a region in a two-dimensional space.

2. How do you solve a double integral?

To solve a double integral, you first need to determine the limits of integration for both variables. Then, you need to set up the integral using the appropriate formula based on the type of region and the function being integrated. Finally, you can evaluate the integral using various techniques such as substitution, integration by parts, or trigonometric identities.

3. What is the difference between a single and double integral?

A single integral involves finding the area under a curve in one-dimensional space, while a double integral involves finding the volume under a surface in two-dimensional space. Additionally, a single integral has only one variable, while a double integral has two variables.

4. Can you explain the concept of a region in double integrals?

In double integrals, a region refers to the area over which the integral is being evaluated. This region is determined by the limits of integration for both variables and can be a rectangle, triangle, circle, or any other shape. The type of region will determine the appropriate formula to use for solving the integral.

5. How can I check if my solution to a double integral is correct?

You can check your solution to a double integral by using various techniques such as graphing the function and the region, using a calculator or computer software, or using known properties of integrals such as the Fundamental Theorem of Calculus. It is also helpful to double-check your calculations and make sure you have used the correct formulas and limits of integration.

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