Mathematics

Problem

Consider a solid whose rectangular cross sections are taken perpendicular to the

x

-axis over the interval

[a,b]

. Each cross section has height

k

times its base width, where

k>0

is constant and the width at

x

f(x)-g(x)

. Explain how the value of

k

affects the volume of the solid when the region between

y=f(x)

and

y=g(x)

remains fixed.

Intuitive explanation

The volume of a solid with rectangular cross sections depends on both the area of each cross section and how that area is distributed. Since each rectangle has width $f(x)-g(x)$ and height $k[f(x)-g(x)]$ , the cross-sectional area is $k[f(x)-g(x)]^2$ —notice the width appears squared. This means volume scales with $k$ linearly: doubling $k$ doubles the volume, because you're stretching all rectangles vertically while keeping their bases fixed.

Intuitive explanation

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