# Thread: Figuring out the bounds of an integral

1. ## Figuring out the bounds of an integral

I understand how to solve double integrals, but I am having trouble figuring out what the bounds are. This is a problem we did in class:

Evaluate the integral (assume S = integral)

SS (4 - x^2 - y^2) dxdy where R is the first quadrant sector of the circle x^2 + y^2 = 4 between the lines y = 0 and x = 0. For reference there is supposed to be an R under the double integral.

The teacher came up with these polar coordinates

0 <= r <= 2 and 0 <= theta <= pi/4

I understand how he got the bounds for r but what about for theta? Thanks in advance

2. Originally Posted by pakman

The teacher came up with these polar coordinates

0 <= r <= 2 and 0 <= theta <= pi/4
I think he made a mistake it should have been,
0<=theta<=pi/2

3. Originally Posted by ThePerfectHacker
I think he made a mistake it should have been,
0<=theta<=pi/2
But how exactly did he derive those bounds? Or you I mean.

4. Originally Posted by pakman
But how exactly did he derive those bounds? Or you I mean.
Because you start at the positive x-axis and move in the positive direction (counterclockwise) on the circle to create a quater-circle. You need to rotate 1/4 of an angle to reach 1/4 of the circle (in the first quadrant) which is pi/2 because a full circle is 2*pi radians.

5. Originally Posted by pakman
I understand how to solve double integrals, but I am having trouble figuring out what the bounds are. This is a problem we did in class:

Evaluate the integral (assume S = integral)

SS (4 - x^2 - y^2) dxdy where R is the first quadrant sector of the circle x^2 + y^2 = 4 between the lines y = 0 and x = 0. For reference there is supposed to be an R under the double integral.

The teacher came up with these polar coordinates

0 <= r <= 2 and 0 <= theta <= pi/4

I understand how he got the bounds for r but what about for theta? Thanks in advance
The integrand is about dx*dy. So the boundaries should be for the dx and the dy. How did dr and d(theta) come into the computations?

No wonder you are lost.

6. Originally Posted by ticbol
The integrand is about dx*dy. So the boundaries should be for the dx and the dy. How did dr and d(theta) come into the computations?

No wonder you are lost.
Sorry, I forgot to mention that the problem was supposed to be converted from dxdy to r dr dtheta.

7. Originally Posted by ThePerfectHacker
Because you start at the positive x-axis and move in the positive direction (counterclockwise) on the circle to create a quater-circle. You need to rotate 1/4 of an angle to reach 1/4 of the circle (in the first quadrant) which is pi/2 because a full circle is 2*pi radians.
Thank you, now I understand (finally!)