Question

Set up the change of variable integral to evaluate ∫∫r x²+y²dA where R is the ellipse given by x-xy+y²=2 and using the transformation (change of variable)

x = √2u-√2/3√v,y=√2u+√2/3v

Answer 1

To evaluate the given integral over the ellipse R, use the given change of variable (transformation) and calculate the Jacobian determinant. Express the integral in terms of u and v, substitute the values, and perform the integral over the new region.

Step-by-step explanation:

To evaluate the integral ∫∫r x²+y²dA over the ellipse R defined by x-xy+y²=2, we can use the change of variable (transformation) x = √2u - (√2/3)√v and y = √2u + (√2/3)v.

To set up the change of variable integral, we need to calculate the Jacobian determinant of the given transformation, denoted as J. In this case, J = (√2/3)√2. The integral can then be expressed as ∫∫r x²+y²dA = ∫∫r(u,v)|J|dudv, where r(u,v) represents the new region of integration in the u-v coordinate system. By substituting the values of x and y in terms of u and v, we can rewrite the integral as ∫∫r(u,v)(4u² + 4v²)√2/3√2dudv. Finally, perform the integral over the region r(u,v) to obtain the final result.