Question

Evaluate ∬ Dˣ²ʸ²dA by using the change of variables given by x=v,y=v/u Here, D is the region in the first quadrant bounded by the parabolas y=

​ˣ² and y=ˣ², and the hyperbolas xy=1 and xy=2.

Answer 1

To evaluate the given double integral using the change of variables, we first need to determine the new region in the uv-plane and find the Jacobian of the transformation. Then, we can evaluate the integral over the transformed region using the change of variables.

Step-by-step explanation:

To evaluate ∬ Dˣ²ʸ²dA using the change of variables x=v, y=v/u, we first need to determine the new region D in the uv-plane. D is defined by the curves y=​ˣ², y=ˣ², xy=1, and xy=2. We can express these curves in terms of u and v as u=v and u=v/(v²). The region D is bounded by v=1, v=2, u=1/v, and u=2/v².

Next, we need to find the Jacobian of the transformation. The Jacobian matrix J is given by:

J = [[∂x/∂u, ∂x/∂v], [∂y/∂u, ∂y/∂v]]

After computing the Jacobian, we can evaluate the integral ∬ Dˣ²ʸ²dA in the uv-plane by transforming it into an integral over D' using the change of variables. The new integral will be ∬ D'g(u,v)dudv, where g(u,v) is the function representing the integrand in terms of u and v.