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
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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