Region is bounded above by the graph of and below by the –axis for

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

Problem

Region

T

is bounded above by the graph of

y=\sin x

and below by the

x

–axis for

0\le x\le \pi

. Consider horizontal slices of

T

; that is, at a fixed

y

with

0\le y\le 1

, the horizontal segment lies between

x=\arcsin y

on the left and

x=\pi-\arcsin y

on the right. Two solids share region

T

as their common base. • Solid P has cross sections taken perpendicular to the

y

–axis that are squares. • Solid Q has cross sections taken perpendicular to the

y

–axis that are rectangles whose height at each

y

is three times the width of the corresponding square cross section of Solid P. Answer the following. Exact values are required; no decimal approximations. (a) Let

w(y)

denote the horizontal width of region

T

at height

y

. Show that

w(y)=\pi-2\arcsin y

for

0\le y\le 1

. (b) Write an explicit definite integral for the volume

V_{P}

of Solid P. (c) Write an explicit definite integral for the volume

V_{Q}

of Solid Q. (d) Evaluate

V_{P}

and

V_{Q}

, then compute

V_{Q}-V_{P}

. Give your final answer in terms of

\pi

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