A planar lamina occupies the region bounded by ,

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

Problem

A planar lamina occupies the region

T

bounded by

y = \sin x

y = 0

, and

x = \pi

. The portion of

T

that lies between

x=0

and

x=\dfrac{\pi}{2}

is removed. The remaining part of

T

is revolved about the

x

–axis to create a hollow solid. (a) On the interval

0 \le x \le \pi

, determine the exact

x

–coordinate at which

y=\sin x

attains its maximum value. Justify your reasoning. (b) Write a single integral representing the volume of the solid obtained by revolving the entire region

T

(before removal) about the

x

–axis. (c) Write a single integral representing the volume of the portion of

T

that is removed, and evaluate this integral. (d) Compute the exact volume of the hollow solid described, showing algebraic simplification at each stage.

Related theory

Optimization
Volume Applications
Applications of Integration

Browse the problem library