Let region be bounded on the left by the –axis

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

Problem

Let region

R

be bounded on the left by the

y

–axis, below by the parabola

y=x^{2}

, and above by the horizontal line

y=4

. The region is revolved about the horizontal line

y=6

to generate a solid of revolution. (1) For

0\le x\le 2

, write an algebraic expression

d_{1}(x)

for the vertical distance between the axis of rotation and the line

y=4

. Simplify your answer. (2) For

0\le x\le 2

, write an algebraic expression

d_{2}(x)

for the vertical distance between the axis of rotation and the curve

y=x^{2}

. Simplify your answer. (3) Explain why

0\le d_{2}(x)\le d_{1}(x)

for all

x

in the interval

[0,2]

. Your explanation should refer to numerical values or inequalities satisfied by the functions involved. (4) Without setting up an integral, state which of the two distances would be subtracted from the other when computing the cross–sectional area of the solid at position

x

, and justify your choice in a sentence.

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