Using a transformation to solve the double integral?

Question

By using the transformation x + y =u, y = uv.

Show that double integral:
 ∫0 to 1∫0 to 1-x  (ey/x+y)dy dx = (e-1)/2

I just need to know what the limits would transform to (in terms of u and v) with an explanation, and then I can take it from there. I ve already found the Jacobian and I know how to express the region in terms of u and v.

Anonymous · Accepted Answer

Please type correctly as use wolfram alpha if needed

int_0^1 int_0^(1 - x) e^(y/(x + y)) dy dx

u = x + y; y = u*v

x = u - u*v; y = u*x

det{{(1 - v),(-u)},{v,u}} = u

now the limits

∫∫ u*(e^v) dA

x = 1 -> u - u*v = 1; u*(1 - v) = 1

x = 0 -> u - u*v = 0; v = 1;

y = 1 - x -> u*v = 1 - (u - u*v) -> u = 1

y = 0 -> u*v = 0

u*(1 - v) = 1; u*v = 0; v = 1; u = 1;

u*(1 - v) = 1; u*v = 0 now you have system of equation as the only possible solution is

u = 1; v = 0

now it is done

int_0^1 int_0^1 (u*(e^v)) dv du

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Using a transformation to solve the double integral?

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