The problem I am trying to solve is that a solid has its basis generator curve in the first quadrant bounded by $y=\sqrt{1-x^2}$ and the coordinate axes. It is noted that the cross sections are perpendicular to the base and are squares.
Upon graphing, I can see that the generator function curve would be a semi-circle repeated in other three quadrants. The solid is what remains after cutting it off by sides of the square.
My problem is that I don't know what method (washer, disc, shell) I would use to find its volume.
If I understood you correctly, washers, discs or shells do not help in solids not formed by revolution. We need to consider all variable square bases of the solid that appears somewhat like:
Please see if the following makes sense. Used symbol $z$ instead of $x$.
$L=L(z)$ is the variable side of square.
Note that $ L(z)=\sqrt 2 y(z),\; y^2+z^2=1 $ at any section.
$$ dV= L^2 dz = 2 y^2 dz=2 (1-z^2) dz,\;L= \sqrt 2 y; $$ $$ V= 2 \int _0^1 (1-z^2) dz =2 (z-\dfrac{z^3}{3})=2(1-\dfrac13)= \dfrac43.$$
Double this for volume of full solid.