Calculer \(\iiint_D^{} f(x,y,z)\,d xd yd z\) :
1
\(D=\{0\le x\le1, 0\le y\le1, 0\le z\le1\}\),
\(f(x,y,z)=\frac1{(x+y+z+1)^3}\).
2
\(D=\{x^2+y^2+z^2 \le R^2\}\),
\(f(x,y,z)=\frac1{\sqrt{a^2-x^2-y^2-z^2}} \quad(a>R>0)\).
3
\(D=\{x\ge0, y\ge0, z\ge0, x+y+z\le1\}\),
\(f(x,y,z)=xyz\).
4
\(D=\{x\ge0, y\ge0, z\ge0, x+y+z\le1\}\),
\(f(x,y,z)=\frac1{(x+y+z+1)^2}\).
5
\(D=\{x^2+y^2\le R^2, 0\le z\le a\}\),
\(f(x,y,z)=x^3+y^3+z^3-3z(x^2+y^2)\).
6
\(D=\{x^2+y^2\le z^2, 0\le z\le 1\}\),
\(f(x,y,z)=\frac z{(x^2+y^2+1)^2}\).
7
\(D=\left\{\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \le 1\right\}\),
\(f(x,y,z)=x^2+y^2\).