# Simplifying an integral by changing the order of integration

Question: Consider a triple integral of the following form $$\int_{x=0}^1 \int_{y=0}^{1} \int_{z=0}^{1} f(x,y,z)dzdydx.$$ Because of the specific $f(\cdot,\cdot,\cdot)$ function I am dealing with, I would like to convert the above integral into the following form $$\int_{x=0}^1 \int_{y=0}^{x} \int_{z=0}^{y} g(x,y,z)dzdydx,$$ where $g(\cdot,\cdot,\cdot)$ is an appropriately defined function. This form is easier to work with because the integrands are nicely ordered as $x\ge y \ge z.$

Example: I am able to do a similar trick for a double integral. Consider $$\int_{x=0}^1 \int_{y=0}^{1} f(x,y)dydx.$$ This integral is equivalent to the sum of two integrals: $$\int_{x=0}^1 \int_{y=0}^{1} f(x,y)dydx=\int_{x=0}^1 \int_{y=0}^{x} f(x,y)dydx+\int_{x=0}^1 \int_{y=x}^{1} f(x,y)dydx.$$ By changing the order of the integration in the last integral above, we obtain $$\int_{x=0}^1 \int_{y=x}^{1} f(x,y)dydx = \int_{y=0}^1 \int_{x=0}^{y} f(x,y)dxdy.$$ Renaming $x$ as $y$ and vice-versa on the integral in the right hand side, we obtain $$\int_{x=0}^1 \int_{y=0}^{x} h(x,y)dydx$$ for an appropriate $h(x,y).$ Thus, the original integral is given as \begin{align} \int_{x=0}^1 \int_{y=0}^{1} f(x,y)dydx &=\int_{x=0}^1 \int_{y=0}^{x} f(x,y)dydx+\int_{x=0}^1 \int_{y=0}^{x} h(x,y)dydx\\ &= \int_{x=0}^1 \int_{y=0}^{x} \left[ f(x,y)+h(x,y) \right] dydx. \end{align}

• This question (particularly the example) bears a lot of similarity to my earlier question: math.stackexchange.com/questions/1403199/… solved by @Brian M. Scott – emper Mar 29 '17 at 6:53
• Why don't you use the same 'trick' on $dz,dy$ then $dy,dx$ respectively ? – Zaid Alyafeai Mar 29 '17 at 10:38
• @ZaidAlyafeai Thanks for the suggestion but that does not really work. In particular, after the first step of changing the order between $z$ and $y,$ I obtain $\int_{x=0}^{1}\int_{y=0}^{1}\int_{z=0}^{y}.$ Applying the trick again on $y$ and $x$ causes me to rename the upper bound on the final integral to $x.$ That defeats the purpose of the exercise to obtain ordered integrands in the form of $x\ge y \ge z.$ – emper Mar 29 '17 at 19:27

We iteratively apply the two-dimensional transformation. In order to do so we recall the slightly more generalized formula \begin{align*} \int_{x=0}^z\int_{y=0}^zf(x,y)\,dx\,dy =\int_{x=0}^z\int_{y=0}^xf(x,y)\,dy\,dx+\int_{y=0}^z\int_{x=0}^yf(x,y)\,dx\,dy\tag{1} \end{align*}

Comment:

• In (2) we apply (1) to the bracketed double integral on the left-hand side.

• In (3) write $g(x,y)=\int_{z=0}^yf(x,y,z)\,dz$ and $h(x,z)=\int_{y=0}^zf(x,y,z)\,dy$

• In (4) we apply again (1) twice, once for the double integral with integrand $g(x,y)$ and once for the double integral with integrand $h(x,z)$.

• In (5) we substitute back for $g(x,y)$ and $h(x,z)$ and observe the bracketed double-integrals need one more transformation according to (1)

• In (6) we finally do this last transformation to the bracketed double integrals.

• @emper: Thanks a lot for accepting my answer and granting the bounty! :-) – Markus Scheuer Apr 10 '17 at 6:40
• Sure, very well deserved! – emper Apr 11 '17 at 4:06