Question

In reading another site I found the following and I don't understand how the limits of integration are derived when the order of integration is changed:

\int f_U(u) h(u)\,du=\int_0^\infty \int_0^\infty\int_{(y+z)/3}^\infty f(3u-y-z,y,z)h(u) 3\,du\,dy\,dz

Changing the order of integration we can write:

\int f_U(u) h(u)\,du=\int_0^\infty\int_0^{3u} \int_0^{3u-z} f(3u-y-z,y,z)h(u) 3\,dy\,dz\,du

Answer 1

Whoops I thought the LaTeX would be compiled the first integral is:

\[\int\limits f_U(u) h(u)\,du=\int\limits_0^\infty \int\limits_0^\infty\int\limits_{(y+z)/3}^\infty f(3u-y-z,y,z)h(u) 3\,du\,dy\,dz\]

Answer 2

And the second integral is:
\[\int\limits f_U(u) h(u)\,du=\int\limits_0^\infty\int\limits_0^{3u} \int\limits_0^{3u-z} f(3u-y-z,y,z)h(u) 3\,dy\,dz\,du\]

Answer 3

I notice the second integration follows the patterns of solving (3u-y-z) for y, then z, and finally u - if you follow the integration signs from right to left (inside out). I'm confused about the pattern of the first integration though.
Its been so long since I've done this that I think my calculator was steam powered!