Volume of Two Perpendicular Cylinders

This interview question considers the solid formed by the intersection of two congruent, perpendicular cylinders whose axes cross at right angles. The candidate is asked to find the volume of the overlapping region, a classic calculus and solid-geometry problem that appears in quantitative interviews for roles requiring strong spatial reasoning and integration skills. The setup highlights symmetry in three dimensions and asks the candidate to translate a geometric picture into a computable quantity, without relying on memorized formulas.

The problem leans heavily on slicing methods and setting up an appropriate integral, often via cross-sections taken along one coordinate axis. It tests the ability to express the boundary of the intersection using equations, derive the shape and dimensions of each slice, and integrate to obtain the total volume. Interviewers watch for recognizing symmetry to simplify the work, choosing a clean coordinate system, handling square roots and bounds correctly, and clearly justifying each step from geometry to algebra to calculus.

What it tests

When two or more solids intersect, the volume of their intersection can often be analyzed by examining the cross-sectional area at each height or position, exploiting symmetry and geometric constraints. The key is to realize that the intersection at a given level is determined by the overlap of the individual cross-sections of each solid at that level. For many symmetric configurations, these cross-sections take on regular shapes (like squares or rectangles) whose dimensions are dictated by the intersection of the original solids' boundaries. This approach reduces a 3D volume problem to a 2D area problem, which can then be integrated over the relevant dimension. The pattern holds because the volume is fundamentally an accumulation of these infinitesimal cross-sectional areas, and symmetry often allows for simplification or recognition of familiar shapes.