Math Help

Change of Variables in Integrals

In a Nut Shell: Sometimes the evaluation of a double integral over a region, R, is

difficult due to the complexity of the integrand, F(x,y), the shape of the area or both.

I = ∫ ∫ F(x,y) dx dy

In such cases, it might prove helpful to change the variables of integration from

(x,y) to (u,v) to simplify either the integrand, the region of integration, or both.

The strategy is to find a transformation, called the Jacobian transformation,

that provides this simplification.

Let dx dy be the original element of area for region R and J_T(u,v) du dv be the

element of area of the transformed region.

Then given sufficient continuity

dx dy = J_T(u,v) du dv

where = J_T(u,v) = the Jacobian transformation defined as follows:

∂x/∂u ∂x∂v

J_T(u,v) = det

∂y/∂u ∂y∂v

and where det denotes the 2x2 determinant of the partial derivatives. The integral

then becomes:

I = ∫ ∫ F(x,y) dx dy = ∫ ∫ F(u,v) J_T(u,v) du dv

R S

R is the original region of integration and S is the transformed region of integration.