Double integral dosen't give same result using Fubini Theorem

Question

I got the equation 0 < y $\le$ x < 1 $\iint k dxdy$ = 1 , where k is a constant but dosen't really matter ... My problem is that I've chose the domains of integration x $\in $ ( 0 , 1 ) and y $\in $ ( 0 , x ) thus I can write the integral in 2 ways : k$\int_0^1\int_0^x 1 dydx = 1$ giveing me k = 2 ... If I do the other way around Fubini Theorem says that I should get the same result but the double integral gives me k*x = 1 thus makeing k= 1 / x ... Why ?

Answer 1

On the one hand,$$\int_0^1\int_0^xdydx=\int_0^1xdx=\frac12.$$On the other hand,$$\int_0^1\int_y^1dxdy=\int_0^1(1-y)dy=\frac12.$$