Let R be the region in the first quadrant bounded by the curve y=x and the x-axis between x=0 and x=9. A three-dimensional solid is constructed so that every cross-section taken perpendicular to the x-axis is a square whose base coincides with the segment lying in R at that value of x.
Calculate the exact volume of the solid.
a. Provide a complete solution, including (i) the algebraic expression for the side length of a typical square cross-section, (ii) the corresponding area function, (iii) an appropriate definite integral, and (iv) the exact value of the volume.