Let be the region in the first quadrant bounded above by the curve and below by the…

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let

R

be the region in the first quadrant bounded above by the curve

y=\sqrt{x}

and below by the

x

–axis for

0\le x\le 4

. A solid is constructed on

R

as its base in the following way: • For

0\le x\le 2

, cross sections taken perpendicular to the

x

–axis are equilateral triangles whose side lies in

R

. • For

2\le x\le 4

, cross sections taken perpendicular to the

x

–axis are semicircles whose diameter lies in

R

. (a) Express the volume

V_1

of the portion of the solid corresponding to

0\le x\le 2

as a definite integral. Do not evaluate yet. (b) Express the volume

V_2

of the portion of the solid corresponding to

2\le x\le 4

as a definite integral. Do not evaluate yet. (c) Evaluate each integral from parts (a) and (b) exactly, and determine the total volume

V=V_1+V_2

of the entire solid. (d) State the units of volume in terms of the units used for

x

and

y

, and briefly interpret the physical meaning of your numerical answer.