need help Find the volume of the resulting solid if the region under the curve10+ 3x + 2from x = 0 to x = 1 is rotated about the x-axis and the y-axis.(a) x-axisy =(b) y-axisx²

Consider the curves , x=0, x=1 about x-axisFor this region of the disk is Volume = ,…

Answered: Find the volume of the resulting solid…

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

need help

Find the volume of the resulting solid if the region under the curve
10
+ 3x + 2
from x = 0 to x = 1 is rotated about the x-axis and the y-axis.
(a) x-axis
y =
(b) y-axis
x²

Expert Solution

Step 1: Introduction to given details

Consider the curves $y space equals space fraction numerator 10 over denominator x squared plus 3 x plus 2 end fraction$ , x=0, x=1 about x-axis

For this region of the disk is $y space equals space fraction numerator 10 over denominator x squared plus 3 x plus 2 end fraction$

Volume = $integral subscript a superscript b straight pi space straight y squared space d x$ ,

$equals integral subscript 0 superscript 1 straight pi space open parentheses fraction numerator 10 over denominator straight x squared plus 3 straight x plus 2 end fraction close parentheses squared space d x comma equals integral subscript 0 superscript 1 straight pi space open parentheses 100 over left parenthesis straight x squared plus 3 straight x plus 2 right parenthesis squared close parentheses space d x comma equals 100 straight pi space integral subscript 0 superscript 1 fraction numerator 1 over denominator left parenthesis straight x plus 1 right parenthesis squared space. space left parenthesis straight x plus 2 right parenthesis squared end fraction space dx equals space 100 straight pi space integral subscript 0 superscript 1 open parentheses fraction numerator 2 over denominator straight x plus 2 end fraction plus 1 over left parenthesis straight x plus 2 right parenthesis squared minus fraction numerator 2 over denominator straight x plus 1 end fraction plus 1 over left parenthesis straight x plus 1 right parenthesis squared close parentheses dx equals space 100 straight pi space open parentheses 2 ln open parentheses 3 close parentheses space minus space 4 ln open parentheses 2 close parentheses plus 2 over 3 close parentheses equals 28.6835$