1. where R: |x| + |y| <= 2

2. A lamina occupies the part of the disk x^2 + y^2 <= 1 in the first quadrant. Find its center of mas if the density at any given point is proportional to the square of its distance from the origin.

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- Jul 16th 2008, 05:14 PMalgebrapro18Double Integration and its Applications
1. where R: |x| + |y| <= 2

2. A lamina occupies the part of the disk x^2 + y^2 <= 1 in the first quadrant. Find its center of mas if the density at any given point is proportional to the square of its distance from the origin. - Jul 16th 2008, 05:38 PMmr fantastic
The region of integration in the xy-plane is bounded by the following four lines:

x + y = 2 when x > 0 and y > 0, that is, the line y = -x + 2 in the first quadrant.

-x + y = 2 when x < 0 and y > 0, that is, the line y = x + 2 in the second quadrant.

-x - y = 2 when x < 0 and y < 0, that is, the line y = -x - 2 in the third quadrant.

x - y = 2 when x > 0 and y < 0, that is, the line y = x - 2 in the fourth quadrant.

The region is therefore a diamond centered on the origin.

Life is much easier if you now make the following coordinate transformation (suggested by this region):

.... (1)

.... (2)

It follows from (1) and (2) that and . Therefore the Jacobian of the transformation is ........

The region of integration in the uv-plane is a square. The integrations are trivial. - Jul 16th 2008, 05:39 PMmr fantastic
- Jul 16th 2008, 06:19 PMalgebrapro18
- Jul 16th 2008, 06:32 PMmr fantastic
- Jul 16th 2008, 06:49 PMalgebrapro18
I really don't feel like tying out all my work but can you check my answers?

M = (pi*k)/8

Center of mass = 8/(5*pi) - Jul 16th 2008, 07:15 PMmr fantastic