Let be a planar region with width at position

MATHEMATICS · Subjective · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let

R

be a planar region with width

w(x)

at position

x

. Two solids are built on

R

using cross sections perpendicular to the

x

-axis. • Solid

M

has square cross sections of side

w(x)

. • Solid

N

has semicircular cross sections whose diameters equal

w(x)

. Determine the exact constant

k

such that

\text{Vol}(M)=k\,\text{Vol}(N)