I do not know the exact location in his Collected Works but Dirichlet found the $n$-volume of
$$ x_1, x_2, \ldots, x_n \geq 0 $$ and $$ x_1^{a_1} + x_2^{a_2} + \ldots + x_n^{a_n} \leq 1 .$$
For example with $n=3$ the volume is $$ \frac{ \Gamma \left( 1 + \frac{1}{a_1} \right) \Gamma \left( 1 + \frac{1}{a_2} \right) \Gamma \left( 1 + \frac{1}{a_3} \right) }{ \Gamma \left(1 + \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} \right) } $$
Note that this has some attractive features. The limit as $ a_n \rightarrow \infty $ is just the expression in dimension $n-1,$ exactly what we want. Also, we quickly get the volume of the positive "orthant" of the unit $n$-ball by setting all $a_j = 2,$ and this immediately gives the volume of the entire unit $n$-ball, abbreviated as $$ \frac{\pi^{n/2}}{(n/2)!} $$
I think he also exactly evaluated the integral of any monomial $$ x_1^{b_1} x_2^{b_2} \cdots x_n^{b_n} $$ on the same set.
So the question would be: given, say, positive integers $a,b,c,$ find the volume of $x,y,z \geq 0$ and $ x^a + y^b + z^c \leq 1.$ If you like, fix the exponents, the triple $a=2, b=3, c=6$ comes up in a book by R.C.Vaughan called "The Hardy-Littlewood Method," page 146 in the second edition, where he assumes the reader knows this calculation.
This came back to mind because of a recent closed question on the area of $x^4 + y^4 \leq 1.$