# Figuring out the bounds of an integral

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• Feb 12th 2007, 07:39 PM
pakman
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
• Feb 12th 2007, 08:13 PM
ThePerfectHacker
Quote:

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
• Feb 12th 2007, 08:25 PM
pakman
Quote:

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.
• Feb 12th 2007, 08:42 PM
ThePerfectHacker
Quote:

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.
• Feb 13th 2007, 12:54 AM
ticbol
Quote:

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.
• Feb 13th 2007, 07:31 AM
pakman
Quote:

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.
• Feb 13th 2007, 07:32 AM
pakman
Quote:

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!)