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.
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.
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.