Can anyone help to calculate the following integral: $$\int_a^\infty \mathrm dy_1\int_a^\infty\mathrm dy_2 \int_a^\infty \mathrm dy_3 \delta\left(y_1+y_2+y_3-3\right)$$ with a<0?
According to maple the result is: $\frac{9}{2}\left(1-a\right)^2$ but I don't know how get this result analytically.
Thanks for helping!
By translating, we can see that are integrating on the projection of the plane
$$x_1+x_2+x_3=3(1-a)$$
onto the $x_1x_2$ plane (this is more easily visualized since we are now confined to the first octant). So the integral evaluates as
$$\int_0^\infty dx_1\int_0^\infty dx_2\int_0^\infty dx_3 \: \delta(x_1+x_2+x_3-3(1-a)) = \int_0^{3(1-a)} dx_1\int_0^{3(1-a)-x_1}dx_2$$
$$ = \int_0^{3(1-a)}3(1-a)-x_1\:dx_1 = \frac{9}{2}(1-a)^2$$