Let R be the region in the first quadrant bounded by the graph of

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

Problem

y=\sqrt{x}

, the

x

-axis, and the line

x=4

. A family of solids is generated when R is revolved about the horizontal line

y=k

, where

k>0

. (a) Describe in words how the distance from a typical point

(x,y)

on the curve

y=\sqrt{x}

to the line

y=k

differs from its distance to the

x

-axis. Express this distance algebraically in terms of

x

and

k

. Do not perform any integration. (b) Explain why the limits of integration used to compute the volume of the solid do not depend on the value of

k

. (c) For

k=4

, write the explicit algebraic form of the squared radius that would appear inside an integral set up to compute the volume. State whether an absolute-value symbol is necessary in this particular case and justify your answer.

Related theory

Optimization
Volume Applications
Applications of Integration

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