Can we write the right hand side as a one integral formula? (A lecture on multivariable mathematics by Theodore Shifrin on YouTube)

Question

I am watching a lecture on multivariable mathematics by Theodore Shifrin on YouTube.

I want to know the answer to this problem.

My answer is the following. My answer uses two integral formulae.

$$\int_{0}^{1}\int_{0}^{1-x}\int_{0}^{x+y} f(x,y,z)\,dz\,dy\,dx=\int_{0}^{1}\int_{0}^{z}\int_{z-x}^{1-x} f(x,y,z)\,dy\,dx\,dz+\int_{0}^{1}\int_{z}^{1}\int_{0}^{1-x} f(x,y,z)\,dy\,dx\,dz.$$

Can we write the right hand side as a one integral formula like this?

$$\int_{0}^{1}\int_{0}^{1-x}\int_{0}^{x+y} f(x,y,z)\,dz\,dy\,dx=\int_{*}^{*}\int_{*}^{*}\int_{*}^{*} f(x,y,z)\,dy\,dx\,dz.$$

Answer 1

I have found the answer in the next lecture.
And my answer was right!