The following problem is on a study guide for an exam I have tomorrow and i\'m s

Question

The following problem is on a study guide for an exam I have tomorrow and i'm stuck on how to do it. Please breakdown as much as possible please..thanks! "Region (A) which is in quadrant 1 in the xy plane. It is bounded by the x-axis, a line (y=2x) and an ellipse x^2 +(y^2)/4 = 1. (part a) The transformation T(x,y)=(x,2y) maps a different region (B), to A. B is bound by the x-axis, the line y=x, and by the unit circle. Calculate the the Jacobian of T. (part b) Using the new coordinates (you got from part a) compute the x and y coordinates of the centroid. (it may be easier to use polar coordinates)"

Explanation / Answer

We will do a change of coordinates:

Let u = x, v = y/2. Then the equation x^2 +(y^2)/4 = 1, becomes u^2 +v^2 = 1, this is, the unit circle. Also, the line y=2x is now v=u.

So this is what part (a) is saying. So B is the region bounded by u=v, and the unit circle u^2 + v^2 =1 (I am using u and v so that is clear the change of variables).

(b) However, anytime we make a change of variables, we have to compute the jacobian and multiply it inside the integral you are trying to solve.. So the jacobian is the determinant of the matrix with entries |du/dx du/ dy \ dv/dx dv/dy|. So the matrix is | 1 0 \ 0 1/2 | , so the determinant is 1/2.

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