Change of Variables in Integrals  (continued)

3.

The challenging part in changing variables from (x,y) to (u,v) is to find the

transformation    x = x(u,v)  and  y = y(u,v) then using this information to

calculate the Jacobian, JT(u,v).

Use the Jacobian to rewrite the transformed integral as follows:

I  =       F(x,y) dA    =          JT(u,v) du dv

R                               S

One  procedure is as follow:

 Step 1 Graph the region, R, defined by  x - f1(x) = C1,  x - f1(x) = C2  and  by  x - f2(x) = D1,   x -  f2(x) = D2. Step 2 Determine the coordinates of the vertices for each corner of the region in R. Step 3 Find the new variables, u and v, by setting  u = f1(x)  and  v  = f2(x). Step 4 Determine the coordinates of the vertices for each corner of the region in S  by substituting the values determined in step 2. Step 5 Plot the new region of integration, S, in the u-v plane.  Two objectives are to simplify the region of integration and to determine the limits of integration in the u-v plane. Step 6 Calculate the Jacobian, JT(u,v) by taking the partial derivatives ∂x/∂u, ∂x/∂v, ∂y/∂u, and ∂y/∂v  . Step 7 Use the Jacobian to evaluate the integral, I, as given above.