How did Eudoxus and Archimedes use the method of exhaustion?

The method of exhaustion was a rigorous logical procedure developed by the classical Greeks to determine the areas and volumes of geometric figures. It functioned by approximating the area or volume of a target shape using a sequence of simpler figures, such as polygons or prisms, whose dimensions were known.12

Eudoxus pioneered the method as a generalization of the theory of proportions. He demonstrated that a target volume—such as that of a pyramid—could be "exhausted" by showing that any value smaller than the target could be exceeded by an internal stack of prisms, while any value larger could be undercut by an external stack. By proving these bounds, he established exact formulas, such as the volume of a pyramid being $\frac{1}{3}Bh$.2

Archimedes became the method's greatest practitioner, applying it to complex problems like the area of a parabolic segment and the volume of a sphere. He famously used it to establish upper and lower bounds for $\pi$ by increasing the number of sides of inscribed and circumscribed polygons. His work effectively anticipated integral calculus without relying on limits or infinitesimals.321