A region R in the first quadrant is bounded on the left by

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

Problem

x = 1

, on the right by

x = 3

, above by

y = \dfrac{6}{x}

, and below by

y = 0

. A parameter

a

(with

a > 0

) is introduced by translating the line of revolution: the region is revolved about the vertical line

x = -a

. (1) Part 1: (a) For general

a > 0

, write an expression for the volume

V(a)

of the resulting solid as a single definite integral in whichever variable of integration gives an integrand that is independent of

a

. Clearly indicate your limits of integration and provide justification for your orientation choice. (2) Part 2: (b) Evaluate the integral from part (a) to obtain a simplified formula for

V(a)

. (3) Part 3: (c) Show that your result from part (b) can be written in the form

V(a) = k\pi(a + c)^2

, where

k

and

c

are constants to be determined. State the values of

k

and

c

. (4) Part 4: (d) Interpret how the translation distance

a

affects the volume of the solid, referring explicitly to your formula from part (c).

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