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Fund= amental Theorem of Calculus
------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Fund= amental Theorem of Calculus
Let I = =3D [a, b] be the interval a < x < b.
------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0015_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0015_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0016.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Fund= amental Theorem of Calculus
Let I = =3D [a, b] be the interval a < x < b.
A regular-partition of I is a division of I into equal = size smaller sub-intervals.
------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0016_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0016_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0017.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Fund= amental Theorem of Calculus
Let I = =3D [a, b] be the interval a < x < b.
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals
------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0017_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0017_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0024.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Fund= amental Theorem of Calculus
Let I = =3D [a, b] be the interval a < x < b.
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals and = Δx be the length of each sub-interval. 
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Fund= amental Theorem of Calculus
Let I = =3D [a, b] be the interval a < x < b.
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals and = Δx be the length of each sub-interval.  (Note saying that n à∞ is the same as saying Δx à 0)
------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0097_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0097_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Fund= amental Theorem of Calculus
Let I = =3D [a, b] be the interval a < x < b.
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals and = Δx be the length of each sub-interval.  (Note saying that n à∞ is the same as saying Δx à 0)
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Fund= amental Theorem of Calculus
Example: {0, 1/3, 2/3, 1} gives a regular partition of = [0, 1] with n =3D 3:<= /div>
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals and = Δx be the length of each sub-interval.  (Note saying that n à∞ is the same as saying Δx à 0)
Let I = =3D [a, b] be the interval a < x < b.
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Fund= amental Theorem of Calculus
Example: {0, 1/3, 2/3, 1} gives a regular partition of = [0, 1] with n =3D 3:<= /div>
Let I = =3D [a, b] be the interval a < x < b.
0
1
1/3
2/3
Δx=3D1/3
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals and = Δx be the length of each sub-interval.  (Note saying that n à∞ is the same as saying Δx à 0)
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Fund= amental Theorem of Calculus
Example: {0, 1/3, 2/3, 1} gives a regular partition of = [0, 1] with n =3D 3:<= /div>
Let I = =3D [a, b] be the interval a < x < b.
0
1
{0, ¼, ½, ¾, 1} gives the regular partition of [0, 1] with n =3D 4:
1/3
2/3
Δx=3D1/3
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals and = Δx be the length of each sub-interval.  (Note saying that n à∞ is the same as saying Δx à 0)
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Fund= amental Theorem of Calculus
Example: {0, 1/3, 2/3, 1} gives a regular partition of = [0, 1] with n =3D 3:<= /div>
Let I = =3D [a, b] be the interval a < x < b.
0
1
{0, ¼, ½, ¾, 1} gives the regular partition of [0, 1] with n =3D 4:
0
1
&fra= c12;
&fra= c34;
&fra= c14;
1/3
2/3
Δx=3D1/3
Δx=3D1/4
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals and = Δx be the length of each sub-interval.  (Note saying that n à∞ is the same as saying Δx à 0)
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Content-Type: image/gif R0lGODlhbwApAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABr ACUAgAAAAAAAAALfhI+py+0Xwpu02otXzDduDoaa+HgfiXJSqnjAycbTKr8fXOcJbePZ6dPpNiZX q2hCwILCGpHHnNGSzSYVwusojdWhL1rCGsDd0NVmvpLLKi7390azY8i1eU6XxPH8vv8PGChosTcI WFdo6ITIaEeYxdYomVgBOZZlR4k1KVnGBGXZ6MCpeCZ2qeGoqPWpZ7kqdBOqOQqLsTT7urOlCosT RHtZZPuY2iI32ks4l4jpquupDKcLNCWdtylTOIx6evh8zcBpfQRdpWb+yN0wG4hLLPibDh/rTP/X Pn9/HuxQAAA7 ------=_NextPart_01C97CE3.AA497110 Content-Location: file:///C:/C86C3F13/1FundamentalTheoremofCalaculus_files/slide0023.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Fund= amental Theorem of Calculus
Example: {0, 1/3, 2/3, 1} gives a regular partition of = [0, 1] with n =3D 3:<= /div>
Let I = =3D [a, b] be the interval a < x < b.
0
1
{0, ¼, ½, ¾, 1} gives the regular partition of [0, 1] with n =3D 4:
0
1
&fra= c12;
&fra= c34;
&fra= c14;
1/3
2/3
Δx=3D1/3
Δx=3D1/4
In general Δx =3D (b – a)/n.
A regular-partition of I is a division of I into equal = size smaller sub-intervals. We will use n to denote the number of the subintervals and = Δx be the length of each sub-interval.  (Note saying that n à∞ is the same as saying Δx à 0)
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Fund= amental Theorem of Calculus
Let y =3D f(x) be a continuous function over [a, = b].
 
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Fund= amental Theorem of Calculus
Let y =3D f(x) be a continuous function over [a, = b].
 
a=3Dx0
b=3Dxn
y =3D f(x)
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Fund= amental Theorem of Calculus
a=3Dx0
b=3Dxn
y =3D f(x)
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a, = b].
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Fund= amental Theorem of Calculus
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a, = b].
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Fund= amental Theorem of Calculus
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x n-1
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a,= b]. We select an arbitrary point     in each sub-interval [x i – 1, xi]. <= /span>
xi
*
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Fund= amental Theorem of Calculus
a=3Dx0
b=3Dxn
x i-1
x i
x1
x2
y =3D f(x)
x1
*
x n-1
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a,= b]. We select an arbitrary point     in each sub-interval [x i – 1, xi]. <= /span>
xi
*
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Fund= amental Theorem of Calculus
a=3Dx0
b=3Dxn
x i-1
x i
x2
*
x1
x2
y =3D f(x)
x1
*
x n-1
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a,= b]. We select an arbitrary point     in each sub-interval [x i – 1, xi]. <= /span>
xi
*
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Fund= amental Theorem of Calculus
a=3Dx0
b=3Dxn
x i-1
x i
x2
*
x1
x2
xi
*
y =3D f(x)
x1
*
x n-1
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a,= b]. We select an arbitrary point     in each sub-interval [x i – 1, xi]. <= /span>
xi
*
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Fund= amental Theorem of Calculus
a=3Dx0
b=3Dxn
x i-1
x i
x2
*
x1
x2
xi
*
y =3D f(x)
The su= m
f(x<= /span>1<= /span>) = 16;x+  
x1
*
is cal= led a Riemann sum.
*
f(x<= /span>2<= /span>) = 16;x+ …  
*
f(x<= /span>n<= /span>) = 16;x =3D  
*
x n-1
i=3D= 1
n
W= 21; f(xi)Δx  
*
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a,= b]. We select an arbitrary point     in each sub-interval [x i – 1, xi]. <= /span>
xi
*
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Fund= amental Theorem of Calculus
a=3Dx0
b=3Dxn
x i-1
x i
x2
*
x1
x2
xi
*
y =3D f(x)
The su= m
f(x<= /span>1<= /span>) = 16;x+  
x1
*
is cal= led a Riemann sum.
*
f(x<= /span>2<= /span>) = 16;x+ …  
*
f(x<= /span>n<= /span>) = 16;x =3D 
*
x n-1
i=3D= 1
n
W= 21; f(xi)Δx  
*
Δx
f(x<= /b>1)
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a,= b]. We select an arbitrary point     in each sub-interval [x i – 1, xi]. <= /span>
xi
*
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Fund= amental Theorem of Calculus
Let y =3D f(x) be a continuous function over [a, = b].
Let the sequence {a=3Dx0= , x1, x<= /span>2= , .. xn=3Db= } be a regular partition of [a,= b]. We select an arbitrary point     in each sub-interval [x i – 1, xi]. <= /span>
a=3Dx0
b=3Dxn
x i-1
xi
*
x i
x2
*
x1
x2
xi
*
y =3D f(x)
The su= m
f(x<= /span>1<= /span>) = 16;x+  
x1
*
is cal= led a Riemann sum.
If f(x) > 0, this sum represents an rectangular approximation of the area between f(x) and the x-= axis.
*
i=3D= 1
n
W= 21; f(xi)Δx  
*
f(x<= /span>2<= /span>) = 16;x+ …  
*
f(x<= /span>n<= /span>) = 16;x =3D 
*
x n-1
Δx
f(x<= /b>1)
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Fund= amental Theorem of Calculus
Fund= amental Theorem of Calculus:
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Fund= amental Theorem of Calculus
Fund= amental Theorem of Calculus:
Let f(x)= be a continuous function over [a, b].
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Fund= amental Theorem of Calculus
Fund= amental Theorem of Calculus:
Let f(x) be a continuous function over [a, b]. The limit <= span style=3D'position:absolute;top:22.25%;left:4.3%;width:94.0%;height:6.0%'><= span style=3D'mso-bidi-font-family:Arial;mso-hansi-font-family:Arial;font-size:= 156%'>of the Riemann sums
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