To find the volume of the solid generated by the surface $z = 16 - x^2 - y^2$ over the given region, we need to carefully set up and evaluate a double integral over the specified region. Here's the step-by-step solution:

1. Understand the Region of Integration

The region is bounded by:

• $y = \sqrt{2}x$ (line 1),
• $y = 4x - 2$ (line 2),
• $x$-axis ($y \geq 0$),
• $y$-axis ($x \geq 0$).

The intersection of $y = \sqrt{2}x$ and $y = 4x - 2$ gives the bounds in $x$ and $y$:

1. Set $\sqrt{2}x = 4x - 2$: $\sqrt{2}x - 4x = -2 \implies x(\sqrt{2} - 4) = -2 \implies x = \frac{2}{4 - \sqrt{2}}.$

   Simplify the fraction: $x = \frac{2(4 + \sqrt{2})}{16 - 2} = \frac{2(4 + \sqrt{2})}{14} = \frac{2 + \sqrt{2}}{7}.$

2. Use this $x$-value to find $y$-coordinates: Substitute $x = \frac{2 + \sqrt{2}}{7}$ into either equation $y = \sqrt{2}x$: $y = \sqrt{2} \cdot \frac{2 + \sqrt{2}}{7} = \frac{2\sqrt{2} + 2}{7}.$

2. Volume Integral

The volume is given by: $V = \int \int_R (16 - x^2 - y^2) \, dA,$ where $R$ is the region bounded by the lines $y = \sqrt{2}x$ and $y = 4x - 2$.

Setting Bounds:

1. $x$-bounds are $0$ to $\frac{2 + \sqrt{2}}{7}$,
2. For a fixed $x$, $y$-bounds are from $y = \sqrt{2}x$ to $y = 4x - 2$.

Thus, the integral becomes: $V = \int_0^{\frac{2 + \sqrt{2}}{7}} \int_{\sqrt{2}x}^{4x - 2} (16 - x^2 - y^2) \, dy \, dx.$

3. Solve the Inner Integral

For the inner integral: $\int_{\sqrt{2}x}^{4x - 2} (16 - x^2 - y^2) \, dy = \int_{\sqrt{2}x}^{4x - 2} 16 \, dy - \int_{\sqrt{2}x}^{4x - 2} x^2 \, dy - \int_{\sqrt{2}x}^{4x - 2} y^2 \, dy.$

1. $\int_{\sqrt{2}x}^{4x - 2} 16 \, dy = 16[y]_{\sqrt{2}x}^{4x - 2} = 16((4x - 2) - (\sqrt{2}x))$.
2. $\int_{\sqrt{2}x}^{4x - 2} x^2 \, dy = x^2[y]_{\sqrt{2}x}^{4x - 2} = x^2((4x - 2) - (\sqrt{2}x))$.
3. $\int_{\sqrt{2}x}^{4x - 2} y^2 \, dy = \left[\frac{y^3}{3}\right]_{\sqrt{2}x}^{4x - 2} = \frac{(4x - 2)^3}{3} - \frac{(\sqrt{2}x)^3}{3}$.

4. Evaluate the Outer Integral

Substitute the results of the inner integrals into the outer integral and simplify. This involves integrating with respect to $x$ over $[0, \frac{2 + \sqrt{2}}{7}]$.

Would you like a fully computed numerical result or further symbolic steps?