Just to convert my comment to an answer.
If you want to use triple integral to find the volume. There are two ways to do this.
First method: Direct triple integral. You have to find the plane equations for all faces of this tetrahedron. Due to the piecewise nature, the limits of the integral are somewhat messy.
Second method: We can transform the points $ A(1,2,3)$, $B(-2,1,5)$, $C(3,7,1)$ to $A'(1,0,0)$, $B'(0,1,0)$, $C'(0,0,1)$. So that the volume of the new tetrahedron is easy to compute. The transformation matrix from $A',B',C'$ to $A,B,C$ is:
$$
T = \begin{pmatrix} 1&-2 &3
\\
2 &1 &7
\\
3 &5 &1\end{pmatrix} .
$$
Because the linearity this is also the Jacobian matrix, so
$$
\mathrm{Volume} = \iiint_{OABC} 1 \,dxdydz = \iiint_{OA'B'C'} 1 \,{|\det (T)|}\,dx'dy'dz' = \frac{17}{2}.
$$
Another tip with a formula to compute $n$-simplex, I took it from my computational geometry notes:
$$
|V| = \frac{1}{3!}\left|\det
\begin{pmatrix}
x_1 & x_2 & x_3 & x_4\\
y_1 & y_2 & y_3 & y_4\\
z_1 & z_2 & z_3 & z_4\\
1 & 1 & 1 & 1\\
\end{pmatrix}
\right|,
$$
where $(x_i,y_i,z_i)$ are the coordinates for the $i$-th vertex. It bears the same form for the area formula of a triangle with three vertices give
$$
|T| = \frac{1}{2!}\left|\det
\begin{pmatrix}
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3 \\
1 & 1 & 1 \\
\end{pmatrix}
\right|.
$$
$y = 2x^2 +2z^2$ and the plane $y=8$ gives:
$8=2x^2 +2z^2$
$4=x^2 +z^2$
This is a circle and gives you limits for $x$ and $z$ obviously it will be polar coordinates problems in xz space.
Limits for y are solved because your surface is going from paraboloid to the $y=8$, so modify your integral bounds... to be something like: $x=rcos(\phi), z=rsin(\phi)$, $y=y$:
$$\int_{0}^{2\pi}d\phi\int_{0}^{2}dr\int_{2r^2}^8 rdy$$
projecting the section of paraboloid and $y=8$ you get xz projection which defines you x and z bounds.
Best Answer
HINT.....Do you need to use calculus? The volume of a tetrahedron with neighbouring edge vectors $\underline{a}, \underline{b}, \underline{c}$ is $$|\frac 16\underline{a}\cdot(\underline{b}\times\underline{c})|$$