A garden sculpture is designed so that when placed upright

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

Problem

A garden sculpture is designed so that when placed upright, its base region in the

xy

-plane is bounded on the right by

x=4

, on the left by

x=0

, above by

y=2\sin\!\left(\frac{\pi x}{4}\right)

, and below by

y=0

. Cross sections perpendicular to the

x

-axis alternate in blocks of one unit of

x

-length: • On the interval

[0,1)

the cross section is a rectangle whose height equals twice its base. • On

[1,2)

the cross section is a square. • On

[2,3)

the cross section is again a rectangle of the same type as on

[0,1)

. • On

[3,4]

the cross section is a square. (a) For

x

[0,4]

, express the base length of a cross section in terms of

x

. (b) Write a piecewise-defined expression for the cross-sectional area

A(x)

on the entire interval

0\le x\le 4

. (c) Construct, in a single expression, the definite integral whose value equals the total volume of the sculpture. (d) Evaluate the integral obtained in part (c) to find the exact volume.

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