Answer
The volume of the solid is $16$.
Work Step by Step
1) The sketch is enclosed below.
2) We already know that the cross-section is a square, whose diagonal runs from $y=-\sqrt x$ to $y=\sqrt x$. As we can see in the sketch, the diagonal equals $\sqrt x+|-\sqrt x|=2\sqrt x$.
Therefore, the side of the cross-sectional square would be $\frac{2\sqrt x}{\sqrt2}=\sqrt{2x}$
Therefore, we can come up with a formula for $A(x)$, which is the area of the cross-sectional square: $$A(x)=(\sqrt{2x})^2=2x$$
3) Limits of integration: The square lies on the plane from $x=0$ to $x=4$.
4) Integrate $A(x)$ to find the volume of the solid: $$V=\int^4_02xdx=\frac{2x^2}{2}\Big]^4_0=x^2\Big]^4_0=4^2-0^2=16$$