Mathematics

Mathematics

Problem

Let RR be a planar region with width w(x)w(x) at position xx. Two solids are built on RR using cross sections perpendicular to the xx-axis. • Solid MM has square cross sections of side w(x)w(x). • Solid NN has semicircular cross sections whose diameters equal w(x)w(x). Determine the exact constant kk such that Vol(M)=k Vol(N)\text{Vol}(M)=k\,\text{Vol}(N).
Intuitive explanation

The key is to set up volume integrals using the cross-sectional area formula for each solid. Solid MM has square cross sections with area [w(x)]2[w(x)]^2, while solid NN has semicircular cross sections with area 12π[w(x)2]2\frac{1}{2}\pi\left[\frac{w(x)}{2}\right]^2 (since the radius is half the diameter). Computing the ratio of their volumes reveals a constant that doesn't depend on the specific shape of RR — only on the geometry of squares versus semicircles.

Intuitive explanation

The key is to set up volume integrals using the cross-sectional area formula for each solid. Solid MM has square cross sections with area [w(x)]2[w(x)]^2, while solid NN has semicircular cross sections with area 12π[w(x)2]2\frac{1}{2}\pi\left[\frac{w(x)}{2}\right]^2 (since the radius is half the diameter). Computing the ratio of their volumes reveals a constant that doesn't depend on the specific shape of RR — only on the geometry of squares versus semicircles.

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