Separate single integral into multiple integrals I can separate an integral like this, (3) parts.
$\int_{1}^{4}x^{2}dx = \int_{1}^{2}x^{2}dx + \int_{2}^{3}x^{2}dx + \int_{3}^{4}x^{2}dx$
I found out this way to separate double integral into three parts.
$\int_{1}^{4}\int_{1}^{4}x^{2}y^{2}dxdy = \int_{1}^{4}\int_{1}^{2}x^{2}y^{2}dxdy + \int_{1}^{4}\int_{2}^{3}x^{2}y^{2}dxdy + \int_{1}^{4}\int_{3}^{4}x^{2}y^{2}dxdy$
But the values of the outer integral are unchanged (1 to 4). Is there any way to split into multiple parts by changing both the inner and outer integral values?
Like, in triple integral.
$\int_{1}^{4}\int_{1}^{4}\int_{1}^{4}x^{2}y^{2}z^{2}\,dx\,dy\,dz = \int_{1}^{4}\int_{1}^{4}\int_{1}^{2}x^{2}y^{2}z^{2}\,dx\,dy\,dz + \int_{1}^{4}\int_{1}^{4}\int_{2}^{3}x^{2}y^{2}z^{2}\,dx\,dy\,dz + \int_{1}^{4}\int_{1}^{4}\int_{3}^{4}x^{2}y^{2}z^{2}\,dx\,dy\,dz$
or
$\int_{1}^{4}\int_{1}^{4}\int_{1}^{4}x^{2}y^{2}z^{2}\,dx\,dy\,dz = \int_{1}^{4}\int_{1}^{2}\int_{1}^{4}x^{2}y^{2}z^{2}\,dx\,dy\,dz + \int_{1}^{4}\int_{2}^{3}\int_{1}^{4}x^{2}y^{2}z^{2}\,dx\,dy\,dz + \int_{1}^{4}\int_{3}^{4}\int_{1}^{4}x^{2}y^{2}z^{2}\,dx\,dy\,dz$
I can only separate using any one integral. I want to separate using by changing all three integral values.
 A: You can only do this only one at a time, but you can do all 3. For example
\begin{align*}
    \int_1^4\int_1^4\int_1^4x^2y^2z^2dxdydz
    & = \int_1^2\int_1^4\int_1^4x^2y^2z^2dxdydz + \int_2^3\int_1^4\int_1^4x^2y^2z^2dxdydz + \int_3^4\int_1^4\int_1^4x^2y^2z^2dxdydz \\
    & = \int_1^2\int_1^2\int_1^4x^2y^2z^2dxdydz + \int_1^2\int_2^3\int_1^4x^2y^2z^2dxdydz + \int_1^2\int_3^4\int_1^4x^2y^2z^2dxdydz \\
    & + \int_2^3\int_1^2\int_1^4x^2y^2z^2dxdydz + \int_2^3\int_2^3\int_1^4x^2y^2z^2dxdydz + \int_2^3\int_3^4\int_1^4x^2y^2z^2dxdydz \\
    & + \int_3^4\int_1^2\int_1^4x^2y^2z^2dxdydz + \int_3^4\int_2^3\int_1^4x^2y^2z^2dxdydz + \int_3^4\int_3^4\int_1^4x^2y^2z^2dxdydz \\
    & = etc...
\end{align*}
A: Yes you may partition the region of integration into subregions and use the additive property of integration.
For example in  $$ \int_{1}^{4}\int_{1}^{4}x^{2}y^{2}dxdy = \int_{1}^{4}\int_{1}^{2}x^{2}y^{2}dxdy + \int_{1}^{4}\int_{2}^{3}x^{2}y^{2}dxdy + \int_{1}^{4}\int_{3}^{4}x^{2}y^{2}dxdy$$
You can go further and have  $$\int_{1}^{4}\int_{1}^{2}x^{2}y^{2}dxdy=\int_{1}^{2}\int_{1}^{2}x^{2}y^{2}dxdy+\int_{2}^{3}\int_{1}^{2}x^{2}y^{2}dxdy+\int_{3}^{4}\int_{1}^{2}x^{2}y^{2}dxdy$$ and so forth.
