Change of Variables

An appropriate choice of co-ordinates facilities the evaluation of a double or a triple integral. By changing the variables, a given integral can be transformed into a simpler integral involving the new variables.

    Change of Variables in a Double Integral
Let the variables, x, y be changed to the new variables u, v by the transformation
        x = ɸ (u, v), y = ψ(u, v)                                … (1)

Here, ɸ(u, v) and ψ(u, v) are continuous and have continuous first order derivatives in some region Ruv in the uv-plane which corresponds to region Rxy in the xy-pane. Then

    ∫_Rxy f(x,y)dx dy= ∫_Ruv f [Φ(u,v) ,ψ(u,v)] |J|du dv                    … (2)

Here,                J = ∂(x, y)/∂ (u, v) (≠0)
is the Jacobian of transformation from (x, y) to (u,v) co-ordinates.

    Change of Variables in a Triple Integral
Let the variables x, y, z be changed to the new variables u, v, w by the transformation
            x = x(u, v, w), y = y(u, v, w)z = z(u, v, w)                    … (1)
Then, ∫∫∫R_xyz f(x, y, z) dx dy dz
= ∫∫∫R_xyz f[x(u, v, w), y(u, v, w), z(u, v, w)] |J| du dv dw                        … (2)

Here,             J = ∂(x, y, z)/∂(u, v, w) (≠0)
Is the Jacobian of transformation from (x, y, z) to (u, v, w) co-ordinates.