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CalculusCalculus10th Edition

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Chapter 8, Problem 6PS
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Finding a Limit The segment BD is the height of Δ

OAB. Let R be the ratio of the area of Δ

DAB to that of the shaded region formed by deleting Δ

OAB from the circular sector

θ lim x → ∞ +

R.

Chapter 8, Problem 6PS, Finding a Limit The segment BD is the height of OAB. Let R be the ratio of the area of DAB to that

To determine

To calculate: The limit value limθ→0+R where R is the ratio of the area of triangle DAB to that of the shaded region formed by deleting triangle OAB from the circular sector subtended by angle θ.

Explanation

Given:

The R be the ratio of the area of triangle DAB to that of the shaded region formed by deleting triangle OAB from the circular sector subtended by angle θ and the segment BD is the height of Triangle OAB refers to figure provided in the question.

Formula used:

The L’Hospital rule is,

limx→∞ND=00

Then,

limx→∞ND=limx→∞dNdD

The area of triangle is,

A=12b⋅h

Where, b is base and h is height.

Calculation:

Consider the provided figure,

The radius of the circle is unitary,

Now from the triangle ODB,

sinθ=BDOB=BD1≈BD

And,

cosθ=ODOB=OD1≈OD

The area of the triangle DAB,

A=12(DA)(BD)=12(OA−OD)(BD)=12(1−cosθ)(sinθ)

Then, the shaded area is,

Shaded area=θ2−12(1)(BD)=θ2−12sinθ

So that, the ratio R is,

R=Area of Triangle DABShaded area=1/2(1−cosθ)sinθ1/2(θ−sinθ)

Now take

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P-15RECh-8 P-16RECh-8 P-17RECh-8 P-18RECh-8 P-19RECh-8 P-20RECh-8 P-21RECh-8 P-22RECh-8 P-23RECh-8 P-24RECh-8 P-25RECh-8 P-26RECh-8 P-27RECh-8 P-28RECh-8 P-29RECh-8 P-30RECh-8 P-31RECh-8 P-32RECh-8 P-33RECh-8 P-34RECh-8 P-35RECh-8 P-36RECh-8 P-37RECh-8 P-38RECh-8 P-39RECh-8 P-40RECh-8 P-41RECh-8 P-42RECh-8 P-43RECh-8 P-44RECh-8 P-45RECh-8 P-46RECh-8 P-47RECh-8 P-48RECh-8 P-49RECh-8 P-50RECh-8 P-51RECh-8 P-52RECh-8 P-53RECh-8 P-54RECh-8 P-55RECh-8 P-56RECh-8 P-57RECh-8 P-58RECh-8 P-59RECh-8 P-60RECh-8 P-61RECh-8 P-62RECh-8 P-63RECh-8 P-64RECh-8 P-65RECh-8 P-66RECh-8 P-67RECh-8 P-68RECh-8 P-69RECh-8 P-70RECh-8 P-71RECh-8 P-72RECh-8 P-73RECh-8 P-74RECh-8 P-75RECh-8 P-76RECh-8 P-77RECh-8 P-78RECh-8 P-79RECh-8 P-80RECh-8 P-81RECh-8 P-82RECh-8 P-83RECh-8 P-84RECh-8 P-85RECh-8 P-86RECh-8 P-87RECh-8 P-88RECh-8 P-89RECh-8 P-90RECh-8 P-91RECh-8 P-1PSCh-8 P-2PSCh-8 P-3PSCh-8 P-4PSCh-8 P-5PSCh-8 P-6PSCh-8 P-7PSCh-8 P-8PSCh-8 P-9PSCh-8 P-10PSCh-8 P-11PSCh-8 P-12PSCh-8 P-13PSCh-8 P-14PSCh-8 P-15PSCh-8 P-16PSCh-8 P-17PSCh-8 P-18PSCh-8 P-19PSCh-8 P-20PSCh-8 P-21PSCh-8 P-22PS

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