Let be a fixed constant

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

Problem

Let

k>1

be a fixed constant. Region

A

is bounded above by

y=kf(x)

, below by the

x

-axis, and on the sides by

x=a

and

x=b

with

0<a<b

. Region

B

is bounded above by

y=f(x)

, below by the

x

-axis, and on the sides by

x=ka

and

x=kb

. Both regions are revolved about the

x

-axis, producing solids with volumes

V_A

and

V_B

, respectively. Assume

f

is continuous and non-negative on the larger of the two intervals. (a) Decide whether

V_A

is less than, equal to, or greater than

V_B

. Support your decision with clear qualitative reasoning rather than explicit calculation. (b) State one specific property of the disc cross-sections that guarantees your answer in part (a) remains valid for every continuous non-negative choice of

f

. (c) Identify a non-trivial function

f

and constants

a,\,b,\,k

(with

k>1

) for which

V_A

and

V_B

are equal. Explain why equality occurs for your example.

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