Mathematics

Problem

Let

R

be the region bounded on the left by the

y

–axis, below by

y=0

, and above by the curve

y = \ln(x+1)

for

0 \le x \le 2

. A solid is created whose cross sections, taken perpendicular to the

y

–axis, are isosceles right triangles with one leg lying in

R

and perpendicular to the

y

–axis. (a) Express the horizontal leg length

\ell(y)

of a cross section in terms of

y

. (b) Derive a simplified formula for the cross–sectional area

A(y)

. (c) Write, with correct limits of integration, a definite integral for the volume of the solid. (d) Evaluate the integral obtained in part (c) to determine the exact volume.

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