Let us regard the standard simplex in two dimensions
$S = \{ x_1 + x_2 \leq 1, x_1 \geq 0, x_2 \geq 0 \}$
Where does the following calculation fail:
$ \int_S x_1^2 = \int_0^1 x_1^2 \int_0^{1-x_1} dx_2 dx_1 = \int x_1^2 ( 1 - x_1 ) dx_1 = \int_0^1 x_1^2 - x_1^3 dx_1 = \frac{1}{3} - \frac{1}{4} = \frac{1}{12}$
while
$ \int_S x_1^2 = \int_0^1 \int_0^{1-x_2} x_1^2 dx_1 dx_2 = \int_0^1 \dfrac{(1-x_2)^3}{3} dx_2 = \frac{1}{3}\sum^3_{i=0}\int_0^1 x_2^i dx_2 = \frac{1}{3}\sum^3_{i=0} \dfrac{1}{i+1} dx_2$
$ = \frac{1}{3}( \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} ) = \frac{7}{36}$
I suppose the error is the error is in the first equality signs (except I have done some very dumb mistake that I am blind to.