Tetrahedron $x + y + z = a$

Here $A (9, 0, 0), B (0, a, 0), C (0, 0, a)$

Volume $=\int\limits^a_{z=0}\int\limits^{a-z}_{y=0}\int\limits^{a-y-z}_{x=0}dxdydz\\ =\int\limits_0^a\int\limits_0^{a-z}[x]_0^{a-y-z}dydz \\ = \int\limits^a_{z=0}\int\limits_{y=0}^{a-z}(a-y-z)dydz\\ = \int\limits_0^a\Bigg[ay-\dfrac {y^2}2-zy\Bigg]_0^{a-z}dz\\ Volume= \int\limits_0^a\Bigg[a(a-z)-\dfrac {(a-z)^2}2-z(a-z)\Bigg]dz\\ = \int\limits_0^a(a^2-2a-\dfrac {a^2}2-az+\dfrac {z^2}2\Bigg)dz \\ =\int\limits_0^a\Bigg[\dfrac {a^2z}2- \dfrac {az^2}2 + \dfrac {z^3}{2\times 3}\Bigg]_0^a\\ =\Bigg[\dfrac {a^3}2-\dfrac {a^3}2+\dfrac {a^3}6\Bigg]-[0]\\ Volume =\dfrac {a^3}6$