Mathematics

Problem

A solid has a base in the first quadrant bounded on the left by

y = x

, on the right by

y = \sqrt{x}

, and above by

x = 1

. Cross sections taken perpendicular to the

x

–axis are semicircles whose diameters extend from the line

y = x

to the curve

y = \sqrt{x}

. (a) For

0 \le x \le 1

, write an expression for the diameter

d(x)

of a typical semicircular cross section. (b) Derive a formula for the cross–sectional area

A(x)

in terms of

x

. Show all reasoning. (c) Compute the volume of the solid by evaluating an appropriate definite integral.