Mathematics

Problem

A solid has a base in the interval

0 \le x \le \dfrac{\pi}{4}

between the curves

y = \cos x

and

y = \sin x

. Cross sections taken perpendicular to the

x

–axis are semicircles whose diameters are three times the vertical distance between the two curves at each

x

. (a) Express the diameter

d(x)

of a cross section as an explicit function of

x

for

0 \le x \le \dfrac{\pi}{4}

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

A(x)

in terms of

x

. (c) Set up and evaluate a definite integral giving the exact volume of the solid. (d) The solid is modified so that the constant "3" in the diameter definition is replaced by a positive parameter

k

. Determine a formula for the resulting volume

V(k)

and discuss the behavior of

V(k)

k\to 0^{+}

and as

k\to \infty

. Provide justification.

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