MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C97CE4.1C7CABD0" This document is a Web archive file. If you are seeing this message, this means your browser or editor doesn't support Web archive files. For more information on the Web archive format, go to http://officeupdate.microsoft.com/office/webarchive.htm ------=_NextPart_01C97CE4.1C7CABD0 Content-Location: file:///C:/2A921913/5ArcLengthandAreaofSurfaces.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1

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Leng= ths of Curves
------=_NextPart_01C97CE4.1C7CABD0 Content-Location: file:///C:/2A921913/5ArcLengthandAreaofSurfaces_files/slide0005.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Leng= ths of Curves
Given f= (x) a differentiable function over [a, b]. 
------=_NextPart_01C97CE4.1C7CABD0 Content-Location: file:///C:/2A921913/5ArcLengthandAreaofSurfaces_files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Leng= ths of Curves
a
Given f= (x) a differentiable function over [a, b]. 
y =3D f(x)
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Leng= ths of Curves
a
b
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
y =3D f(x)
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a
b
y =3D f(x)
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x<= /span>1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x<= /span>1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let p0, p1, p2, ..= ,pn=  be = the points on the curve corresp= onding to the x's.  
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x<= /span>1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let p0, p1, p2, ..= ,pn=  be = the points on the curve corresp= onding to the x's.  
p0
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x<= /span>1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let p0, p1, p2, ..= ,pn=  be = the points on the curve corresp= onding to the x's.  
p0
p1
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x<= /span>1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let p0, p1, p2, ..= ,pn=  be = the points on the curve corresp= onding to the x's.  
p0
p1
p2
pi-1
pi
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x<= /span>1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let p0, p1, p2, ..= ,pn=  be = the points on the curve corresp= onding to the x's.  
p0
p1
p2
pi-1
pi
=3D (xi-1, f(xi-1))
The coordinate of pi-1 =3D (xi-1, f(xi-1))=
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x<= /span>1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let p0, p1, p2, ..= ,pn=  be = the points on the curve corresp= onding to the x's.  
p0
p1
p2
pi-1
pi
=3D (xi, f(xi))
=3D (xi-1, f(xi-1))
The coordinate of pi-1 =3D (xi-1, f(xi-1))= and pi=  =3D= (xi= , f(xi-1))= . 
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Leng= ths of Curves
Given f(x) a differentiable function over [a, b].=  
We want to calculate the length of the curve y = =3D f(x).
Let the sequence {a=3Dx0, x<= /span>1= , x2, ..= xn= =3Db} be a regular partition of [a, b].  <= /span>
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let p0, p1, p2, ..= ,pn=  be = the points on the curve corresp= onding to the x's.  
p0
p1
p2
pi-1
pi
The coordinate of pi-1 =3D (xi-1, f(xi-1))= and pi=  =3D= (xi= , f(xi-1))= . 
=3D (xi, f(xi))
The length of the line segment from of pi= -1 to = pi=  = 3;
is Ö(x<= /span>i=  = 211; xi= -1)2 + (f(xi) – f(xi-1))2 =3D Ö(Δx)2 + (f(xi) – f(xi-1)2 
=3D (xi-1, f(xi-1))
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Leng= ths of Curves
The su= m of all the line segments is
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
p0
p1
p2
pi-1
=3D (xi-1, f(xi-1))
p0<= /span>p1 + p1p2 + ... + pn-1pn
pi
=3D (xi, f(xi))
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Leng= ths of Curves
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
p0
p1
p2
pi-1
Ö(Δx)2 + (f(x1) – f(x0))2 + 
Ö(Δx)2 + (f(x2) – f(x1))2 + .. + 
Ö(Δx)2 + (f(xn) – f(xn-1))2  
=3D (xi-1, f(xi-1))
p0<= /span>p1 + p1p2 + ... + pn-1pn
=3D <= /div>
The su= m of all the line segments is
pi
=3D (xi, f(xi))
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Leng= ths of Curves
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
=3D Σ
p0
p1
p2
pi-1
Ö(Δx)2 + (f(x1) – f(x0))2 + 
Ö(Δx)2 + (f(x2) – f(x1))2 + .. + 
Ö(Δx)2 + (f(xn) – f(xn-1))2  
i=3D= 1
n
Ö(Δx)2 + (f(xi) – f(xi-1))2  
=3D (xi-1, f(xi-1))
approx= imates the curve length.
p0<= /span>p1 + p1p2 + ... + pn-1pn
=3D <= /div>
The su= m of all the line segments is
pi
=3D (xi, f(xi))
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R0lGODlhJwAnAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAl ACUAgAAAAP8AAAJQhI+pu8EPWZixwumsTnh7031bKFpkGZ3oo64N5V5wLM200t6grR+5/rsFacNY 0XVcJVHLUlP0/EQ90xGvB6hqtKZrj1sBp7xAstBMRAuxiwIAOw== ------=_NextPart_01C97CE4.1C7CABD0 Content-Location: file:///C:/2A921913/5ArcLengthandAreaofSurfaces_files/slide0057.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Leng= ths of Curves
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
=3D Σ
p0
p1
p2
pi-1
Ö(Δx)2 + (f(x1) – f(x0))2 + 
Ö(Δx)2 + (f(x2) – f(x1))2 + .. + 
Ö(Δx)2 + (f(xn) – f(xn-1))2  
i=3D= 1
n
Ö(Δx)2 + (f(xi) – f(xi-1))2  
=3D (xi-1, f(xi-1))
approx= imates the curve length.
=
p0<= /span>p1 + p1p2 + ... + pn-1pn
=3D <= /div>
Take t= he Δx out of the root: <= /div>
The su= m of all the line segments is
pi
=3D (xi, f(xi))
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------=_NextPart_01C97CE4.1C7CABD0 Content-Location: file:///C:/2A921913/5ArcLengthandAreaofSurfaces_files/slide0057_image127.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwAnAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAl ACUAgAAAAP8AAAJQhI+pu8EPWZixwumsTnh7031bKFpkGZ3oo64N5V5wLM200t6grR+5/rsFacNY 0XVcJVHLUlP0/EQ90xGvB6hqtKZrj1sBp7xAstBMRAuxiwIAOw== ------=_NextPart_01C97CE4.1C7CABD0 Content-Location: file:///C:/2A921913/5ArcLengthandAreaofSurfaces_files/slide0021.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Leng= ths of Curves
= 31;
i=3D1=
n
Ö(Δx)2 + (f(xi) – f(xi-1))2  
=3D <= /span>Σ
i= =3D1
n
ÖΔx2 [1 +        = ;             ]   
(f(xi) ̵= 1; f(xi-1))2
Δ<= b>x2
=3D Σ
Ö(Δx)2 + (f(x1) – f(x0))2 + 
Ö(Δx)2 + (f(x2) – f(x1))2 + .. + 
Ö(Δx)2 + (f(xn) – f(xn-1))2  
i=3D= 1
n
Ö(Δx)2 + (f(xi) – f(xi-1))2  
approx= imates the curve length.
p0<= /span>p1 + p1p2 + ... + pn-1pn
=3D <= /div>
Take t= he Δx out of the root: <= /div>
The su= m of all the line segments is
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Leng= ths of Curves
= 31;
i=3D1=
n
Ö(Δx)2 + (f(xi) – f(xi-1))2  
=3D <= /span>Σ
i= =3D1
n
ÖΔx2 [1 +        = ;             ]   
(f(xi) ̵= 1; f(xi-1))2
Δ<= b>x2
=3D Σ
i=3D= 1
n
Ö1 + [        = ;          ]2  Δx
 f(xi) – f(xi-1)
<= b>Δx
=3D Σ
Ö(Δx)2 + (f(x1) – f(x0))2 + 
Ö(Δx)2 + (f(x2) – f(x1))2 + .. + 
Ö(Δx)2 + (f(xn) – f(xn-1))2  
i=3D= 1
n
Ö(Δx)2 + (f(xi) – f(xi-1))2  
approx= imates the curve length.
p0<= /span>p1 + p1p2 + ... + pn-1pn
=3D <= /div>
Take t= he Δx out of the root: <= /div>
The su= m of all the line segments is
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