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The region bounded by y = 5 and y = x+(4/x) is rotated about the line x=−1. Find the volume of the resulting solid by any method.

 Feb 17, 2017

Best Answer 

 #2
avatar+26398 
+15

The region bounded by y = 5 and y = x+(4/x) is rotated about the line x=−1.

Find the volume of the resulting solid by any method.

 

Shell method:

f(x)=x+4xg(x)=5

 

Rotation around Vertical line: xaxis of rotation=1

Radius of Rotation: r=xxaxis of rotation=x(1)=x+1

Circumference of the cylinder =2πr

Shell Volume =2πr[g(x)f(x)] dx

 

Limits of integration:

f(x)=g(x)x+4x=5|xx2+4=5x|5xx25x+4=0(x1)(x4)=0a=1andb=4

 

Volume:

V=ba2πr[g(x)f(x)] dx=2πbar[g(x)f(x)] dx

V=2π41(x+1)[5(x+4x)] dx=2π41[x(5x4x)+5x4x] dx=2π41(5xx24+5x4x) dx=2π41(4xx2+14x) dx=2π[ 2x2x33+x4ln(x) ]41=2π[ 2(4)2(4)33+(4)4ln(4)(2(1)2(1)33+(1)4ln(1))]=2π[32643+44ln(4)(213+1+0)]=2π[32643+44ln(4)2+131]=2π[336334ln(4)]=2π[33214ln(22)]=2π[1242ln(2)]=2π[128ln(2)]=2π6.45482255552=40.5568462413

 

 

The volume of the resulting solid is 40.5568462413

 

laugh

 Feb 17, 2017
 #1
avatar+130493 
0

The shell method seems logical here.....we have

 

      4

2 pi ∫  (x + 1)[ 5 - (x + 4/x)]  dx  =

      1

 

      4

2 pi ∫   5x - x^2 - 4 + 5 - x - 4/x     dx

      1

 

 

       4

2 pi ∫   4x - x^2 + 1- 4/x     dx    =  [16137 / 1250]  pi units^3

      1

 

 

 

cool cool cool

 Feb 17, 2017
 #2
avatar+26398 
+15
Best Answer

The region bounded by y = 5 and y = x+(4/x) is rotated about the line x=−1.

Find the volume of the resulting solid by any method.

 

Shell method:

f(x)=x+4xg(x)=5

 

Rotation around Vertical line: xaxis of rotation=1

Radius of Rotation: r=xxaxis of rotation=x(1)=x+1

Circumference of the cylinder =2πr

Shell Volume =2πr[g(x)f(x)] dx

 

Limits of integration:

f(x)=g(x)x+4x=5|xx2+4=5x|5xx25x+4=0(x1)(x4)=0a=1andb=4

 

Volume:

V=ba2πr[g(x)f(x)] dx=2πbar[g(x)f(x)] dx

V=2π41(x+1)[5(x+4x)] dx=2π41[x(5x4x)+5x4x] dx=2π41(5xx24+5x4x) dx=2π41(4xx2+14x) dx=2π[ 2x2x33+x4ln(x) ]41=2π[ 2(4)2(4)33+(4)4ln(4)(2(1)2(1)33+(1)4ln(1))]=2π[32643+44ln(4)(213+1+0)]=2π[32643+44ln(4)2+131]=2π[336334ln(4)]=2π[33214ln(22)]=2π[1242ln(2)]=2π[128ln(2)]=2π6.45482255552=40.5568462413

 

 

The volume of the resulting solid is 40.5568462413

 

laugh

heureka Feb 17, 2017

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