Let's consider the general problem of determining the volume $$v(K)$$ of the region where $$\sum f_i(x_i) < K$$; that is, of determining $$\int_{\sum f_i(x_i) < K} \prod dx_i$$. Let us define new variables $$y_i = f_i(x_i)$$; the problem is now expressed as $$\int_{\sum y_i < K} \prod d g_i(y_i)$$, where $$g_i$$ is the inverse of $$f_i$$. In other words, the derivative of $$v$$ is the convolution of the derivatives of the $$g_i$$. Let us refer to this sort of relationship by saying $$v$$ is the "donvolution" of the $$g_i$$.

For convenience, we may also suppose that all the $$g_i$$ take input zero to output zero, and in general act on only nonnegative inputs and outputs, and thus so does $$v$$.

Note that as convolution is commutative, associative, and linear in each argument, so is donvolution.

Let us now consider the specific case where $$g_i(x) = x^{p_i}$$. Our goal, then, is to understand donvolution of power functions.

Note also that convolving a function with the constantly $$1$$ function (restricted to nonnegative inputs) is as good as integrating it from a starting point of $$0$$; this means donvolving a function with the 1th power function, i.e. the identity function, is as good as integrating it (from a starting point of $$0$$).

[TODO: Rewrite the following. Note how the general observation that $$x^n$$ donvolved with $$x^m$$ is some scalar times $$x^{n + m}$$ allows us to derive SOME function $$n!$$ such that $$x^n/n!$$ donvolved with $$x^m/m! = x^{n + m}!/(n + m)!$$, unique up to multiplication by an exponential function, since we can calculate this function's second multiplicative differential and its value at $$0$$. It's clear that $$0! = 1$$, and the power rule of integration tells us that $$n!/[(n - 1)! 1!] = n$$; more generally, we find the asymptotics that $$(N + x)!/N! \sim (kN)^x$$, for some base of exponentiation $$k$$ such that $$k^1 = 1!$$. Since we have the degree of freedom to change exponential factors, we can set $$k$$ to 1 (including in the sense that its fractional powers are all 1), and now the factorial is uniquely defined, and matches the factorial we would get from https://howsridharthinks.wordpress.com/2019/11/19/difference-equations-infinite-sums-generalized-factorial-zeta-functions-etc/ .]

Thus, the $$n$$-fold donvolution of $$x$$ with itself is the result of integrating the constantly 1 function $$n$$ times, $$x^n/n!$$ [that this is the $$n$$-fold integral of the constantly 1 function corresponds to the basic power rule that $$x^n$$ integrates to $$x^{n + 1}/(n + 1)$$]. Which means $$x^n/n!$$ donvolved with $$x^m/m!$$ must be $$x^{n + m}/(n! m!)$$. And more generally, the donvolution of various $$x^{p_i}/p_i!$$ will be $$x^{\sum p_i}/(\sum p_i)!$$.

Which, put another way, tells us that donvolution of various $$x^{p_i}$$ will be $$x^{\sum p_i}$$ divided by the multinomial coefficient $$(\sum p_i)!/(\prod (p_i!))$$. This gives us our $$v(x)$$, completing our desired result.