Let region be the set of points in the -plane that satisfy

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

Problem

Let region

R

be the set of points in the

xy

-plane that satisfy

|x| \leq y \leq 4

. A solid has

R

as its base. For every

y

with

0 \leq y \leq 4

, a cross section perpendicular to the

y

-axis is erected. • For

0 \leq y \leq 2

, each cross section is a square. • For

2 \leq y \leq 4

, each cross section is a rectangle whose height is three times the length of its base. Complete the tasks below. Clearly communicate your reasoning. (1) Part 1: (a) Explain why the width

w(y)

of the base segment at a fixed value of

y

2y

. (2) Part 2: (b) Derive an explicit expression for the area

A_1(y)

of a square cross section when

0 \leq y \leq 2

. (3) Part 3: (c) Write, then evaluate, a definite integral that gives the volume

V_1

contributed by the square cross sections. (4) Part 4: (d) Derive an explicit expression for the area

A_2(y)

of a rectangular cross section when

2 \leq y \leq 4

. (5) Part 5: (e) Write, then evaluate, a definite integral that gives the volume

V_2

contributed by the rectangular cross sections. State the total volume of the solid, using proper units.

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