Consider the solid object that is obtained when the function:y = 6 (cos(x) - 5)is rotated by 2π radians about the x-axis between the limits x =Find the volume of this object.V=Enter your answer to 3 decimal places.9π and x = 11π

Answered: Consider the solid object that is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the solid object that is obtained when the function:
y = 6 (cos(x) - 5)
is rotated by 2π radians about the x-axis between the limits x =
Find the volume of this object.
V
=
Enter your answer to 3 decimal places.
9π and x = 11π

Expert Solution

Step 1: Introduction:

Given: The function $y equals 6 left parenthesis cos left parenthesis x right parenthesis minus 5 right parenthesis$ .

To find the volume of the solid obtained when the function is rotated about the x-axis between the limits $x equals 9 pi$ and $x equals 11 pi$ .

Concept Used:

The volume of the solid formed by revolving $y equals f left parenthesis x right parenthesis$ about the $x$ -axis between $x equals a$ and $x equals b$ is $V equals pi integral subscript a superscript b left square bracket f left parenthesis x right parenthesis right square bracket squared d x$ .

Algebraic Identity: $left parenthesis a plus b right parenthesis squared equals a squared plus b squared plus 2 a b$

Trigonometric Identities : $cos squared left parenthesis x right parenthesis equals fraction numerator 1 plus cos left parenthesis 2 x right parenthesis over denominator 2 end fraction$

$sin left parenthesis n straight pi right parenthesis equals 0$ for any natural number $n$

Integration formula : $integral cos left parenthesis x right parenthesis d x equals sin left parenthesis x right parenthesis$

$integral x to the power of n equals fraction numerator x to the power of n plus 1 end exponent over denominator n plus 1 end fraction$ .