lecture 11 - section 8.7 numerical integrationjiwenhe/math1432/lectures/lecture11.pdf · jiwen he,...
TRANSCRIPT
![Page 1: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/1.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic Approximations
Lecture 11Section 8.7 Numerical Integration
Jiwen He
Department of Mathematics, University of Houston
[email protected]://math.uh.edu/∼jiwenhe/Math1432
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 1 / 14
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Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Area Problem
Partition of [a, b]
Take a partition P = x0, x1, · · · , xn of [a, b]. Then P splits upthe interval [a, b] into a finite number of subintervals [x0, x1], · · · ,[xn−1, xn] with a = x0 < x1 < · · · < xn = b. We have
[a, b] = [x0, x1] ∪ · · · ∪ [xi−1, xi ] ∪ · · · ∪ [xn−1, xn]
Remark
This breaks up the region Ω into n subregions Ω1,· · · ,Ωn:Ω = Ω1 ∪ · · · ∪ Ωi ∪ · · · ∪ Ωn
We can estimate the total area of Ω by estimating the area of eachsubregion Ωi and adding up the results.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 2 / 14
![Page 3: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/3.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Area Problem
Partition of [a, b]
Take a partition P = x0, x1, · · · , xn of [a, b]. Then P splits upthe interval [a, b] into a finite number of subintervals [x0, x1], · · · ,[xn−1, xn] with a = x0 < x1 < · · · < xn = b. We have
[a, b] = [x0, x1] ∪ · · · ∪ [xi−1, xi ] ∪ · · · ∪ [xn−1, xn]
Remark
This breaks up the region Ω into n subregions Ω1,· · · ,Ωn:Ω = Ω1 ∪ · · · ∪ Ωi ∪ · · · ∪ Ωn
We can estimate the total area of Ω by estimating the area of eachsubregion Ωi and adding up the results.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 2 / 14
![Page 4: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/4.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Lower and Upper Sums
Let ∆xi = xi − xi−1, mi = minx∈[xi−1,xi ]
f (x), Mi = maxx∈[xi−1,xi ]
f (x)
mi∆xi = area of ri ≤ area of Ωi ≤ area of Ri = Mi∆xi
Lf (P) :=n∑
i=1
mi∆xi ≤ I =
∫ b
af (x) dx ≤
n∑i=1
Mi∆xi =: Uf (P)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 3 / 14
![Page 5: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/5.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Lower and Upper Sums
Let ∆xi = xi − xi−1, mi = minx∈[xi−1,xi ]
f (x), Mi = maxx∈[xi−1,xi ]
f (x)
mi∆xi = area of ri ≤ area of Ωi ≤ area of Ri = Mi∆xi
Lf (P) :=n∑
i=1
mi∆xi ≤ I =
∫ b
af (x) dx ≤
n∑i=1
Mi∆xi =: Uf (P)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 3 / 14
![Page 6: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/6.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Lower and Upper Sums
Let ∆xi = xi − xi−1, mi = minx∈[xi−1,xi ]
f (x), Mi = maxx∈[xi−1,xi ]
f (x)
mi∆xi = area of ri ≤ area of Ωi ≤ area of Ri = Mi∆xi
Lf (P) :=n∑
i=1
mi∆xi ≤ I =
∫ b
af (x) dx ≤
n∑i=1
Mi∆xi =: Uf (P)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 3 / 14
![Page 7: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/7.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Riemann Sum
Riemann Sum
S∗(P) = f (x∗1 )∆x1 + f (x∗2 )∆x2 + · · ·+ f (x∗n )∆xn
where x∗i is any point picked in [xi−1, xi ] for i = 1, · · · , n. Wehave
Lf (P) :=n∑
i=1
mi∆xi ≤ S∗(P) :=n∑
i=1
f (x∗i )∆xi ≤n∑
i=1
Mi∆xi =: Uf (P)
Limit of Riemann Sums∫ b
af (x) dx = lim
|P|→0[f (x∗1 )∆x1 + f (x∗2 )∆x2 + · · ·+ f (x∗n )∆xn]
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 4 / 14
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Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Riemann Sum
Riemann Sum
S∗(P) = f (x∗1 )∆x1 + f (x∗2 )∆x2 + · · ·+ f (x∗n )∆xn
where x∗i is any point picked in [xi−1, xi ] for i = 1, · · · , n. Wehave
Lf (P) :=n∑
i=1
mi∆xi ≤ S∗(P) :=n∑
i=1
f (x∗i )∆xi ≤n∑
i=1
Mi∆xi =: Uf (P)
Limit of Riemann Sums∫ b
af (x) dx = lim
|P|→0[f (x∗1 )∆x1 + f (x∗2 )∆x2 + · · ·+ f (x∗n )∆xn]
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 4 / 14
![Page 9: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/9.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Riemann Sum
Riemann Sum
S∗(P) = f (x∗1 )∆x1 + f (x∗2 )∆x2 + · · ·+ f (x∗n )∆xn
where x∗i is any point picked in [xi−1, xi ] for i = 1, · · · , n. Wehave
Lf (P) :=n∑
i=1
mi∆xi ≤ S∗(P) :=n∑
i=1
f (x∗i )∆xi ≤n∑
i=1
Mi∆xi =: Uf (P)
Limit of Riemann Sums∫ b
af (x) dx = lim
|P|→0[f (x∗1 )∆x1 + f (x∗2 )∆x2 + · · ·+ f (x∗n )∆xn]
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 4 / 14
![Page 10: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/10.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsArea Problem Lower and Upper Sums Riemann Sum
Riemann Sum
Riemann Sum
S∗(P) = f (x∗1 )∆x1 + f (x∗2 )∆x2 + · · ·+ f (x∗n )∆xn
where x∗i is any point picked in [xi−1, xi ] for i = 1, · · · , n. Wehave
Lf (P) :=n∑
i=1
mi∆xi ≤ S∗(P) :=n∑
i=1
f (x∗i )∆xi ≤n∑
i=1
Mi∆xi =: Uf (P)
Limit of Riemann Sums∫ b
af (x) dx = lim
|P|→0[f (x∗1 )∆x1 + f (x∗2 )∆x2 + · · ·+ f (x∗n )∆xn]
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 4 / 14
![Page 11: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/11.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Regular Partition
Problem
Find the approximate value of
ln 2 =
∫ 2
1
dx
xusing only the values of f (x) = 1
x at
1,6
5,
7
5,
8
5,
9
5, 2.
Numerical Integration
Approximate∫ ba f (x) dx using only values of f at n + 1
equally-spaced points between a and b.
Regular Partition of [a, b]
Let xi = a + i∆x , i = 0, 1, · · · , n, where ∆x = b−an . Then
[a, b] = [x0, x1] ∪ · · · ∪ [xi−1, xi ] ∪ · · · ∪ [xn−1, xn]
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 5 / 14
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Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Regular Partition
Problem
Find the approximate value of
ln 2 =
∫ 2
1
dx
xusing only the values of f (x) = 1
x at
1,6
5,
7
5,
8
5,
9
5, 2.
Numerical Integration
Approximate∫ ba f (x) dx using only values of f at n + 1
equally-spaced points between a and b.
Regular Partition of [a, b]
Let xi = a + i∆x , i = 0, 1, · · · , n, where ∆x = b−an . Then
[a, b] = [x0, x1] ∪ · · · ∪ [xi−1, xi ] ∪ · · · ∪ [xn−1, xn]
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 5 / 14
![Page 13: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/13.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Regular Partition
Problem
Find the approximate value of
ln 2 =
∫ 2
1
dx
xusing only the values of f (x) = 1
x at
1,6
5,
7
5,
8
5,
9
5, 2.
Numerical Integration
Approximate∫ ba f (x) dx using only values of f at n + 1
equally-spaced points between a and b.
Regular Partition of [a, b]
Let xi = a + i∆x , i = 0, 1, · · · , n, where ∆x = b−an . Then
[a, b] = [x0, x1] ∪ · · · ∪ [xi−1, xi ] ∪ · · · ∪ [xn−1, xn]
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 5 / 14
![Page 14: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/14.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Approximations
∫ xi
xi−1f (x) dx f (xi−1)∆x f (xi )∆x f
(xi−1+xi
2
)∆x
Rectangle Approximations of∫ xi
xi−1f (x) dx
f (xi−1)∆x , f (xi )∆x , f(
xi−1+xi
2
)∆x
left-endpoints right-endpoints midpoints
Rectangle Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx
n∑i=1
f (xi−1)∆x ,
n∑i=1
f (xi )∆x ,
n∑i=1
f
(xi−1 + xi
2
)∆x
left-endpoints right-endpoints midpoints
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 6 / 14
![Page 15: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/15.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Approximations
∫ xi
xi−1f (x) dx f (xi−1)∆x f (xi )∆x f
(xi−1+xi
2
)∆x
Rectangle Approximations of∫ xi
xi−1f (x) dx
f (xi−1)∆x , f (xi )∆x , f(
xi−1+xi
2
)∆x
left-endpoints right-endpoints midpoints
Rectangle Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx
n∑i=1
f (xi−1)∆x ,
n∑i=1
f (xi )∆x ,
n∑i=1
f
(xi−1 + xi
2
)∆x
left-endpoints right-endpoints midpoints
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 6 / 14
![Page 16: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/16.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Approximations
∫ xi
xi−1f (x) dx f (xi−1)∆x f (xi )∆x f
(xi−1+xi
2
)∆x
Rectangle Approximations of∫ xi
xi−1f (x) dx
f (xi−1)∆x , f (xi )∆x , f(
xi−1+xi
2
)∆x
left-endpoints right-endpoints midpoints
Rectangle Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx
n∑i=1
f (xi−1)∆x ,
n∑i=1
f (xi )∆x ,
n∑i=1
f
(xi−1 + xi
2
)∆x
left-endpoints right-endpoints midpoints
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 6 / 14
![Page 17: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/17.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Approximations
∫ xi
xi−1f (x) dx f (xi−1)∆x f (xi )∆x f
(xi−1+xi
2
)∆x
Rectangle Approximations of∫ xi
xi−1f (x) dx
f (xi−1)∆x , f (xi )∆x , f(
xi−1+xi
2
)∆x
left-endpoints right-endpoints midpoints
Rectangle Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx
n∑i=1
f (xi−1)∆x ,
n∑i=1
f (xi )∆x ,
n∑i=1
f
(xi−1 + xi
2
)∆x
left-endpoints right-endpoints midpoints
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 6 / 14
![Page 18: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/18.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Approximations
∫ xi
xi−1f (x) dx f (xi−1)∆x f (xi )∆x f
(xi−1+xi
2
)∆x
Rectangle Approximations of∫ xi
xi−1f (x) dx
f (xi−1)∆x , f (xi )∆x , f(
xi−1+xi
2
)∆x
left-endpoints right-endpoints midpoints
Rectangle Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx
n∑i=1
f (xi−1)∆x ,
n∑i=1
f (xi )∆x ,
n∑i=1
f
(xi−1 + xi
2
)∆x
left-endpoints right-endpoints midpoints
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 6 / 14
![Page 19: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/19.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Approximations
∫ xi
xi−1f (x) dx f (xi−1)∆x f (xi )∆x f
(xi−1+xi
2
)∆x
Rectangle Approximations of∫ xi
xi−1f (x) dx
f (xi−1)∆x , f (xi )∆x , f(
xi−1+xi
2
)∆x
left-endpoints right-endpoints midpoints
Rectangle Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx
n∑i=1
f (xi−1)∆x ,
n∑i=1
f (xi )∆x ,
n∑i=1
f
(xi−1 + xi
2
)∆x
left-endpoints right-endpoints midpoints
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 6 / 14
![Page 20: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/20.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Approximations
∫ xi
xi−1f (x) dx f (xi−1)∆x f (xi )∆x f
(xi−1+xi
2
)∆x
Rectangle Approximations of∫ xi
xi−1f (x) dx
f (xi−1)∆x , f (xi )∆x , f(
xi−1+xi
2
)∆x
left-endpoints right-endpoints midpoints
Rectangle Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx
n∑i=1
f (xi−1)∆x ,
n∑i=1
f (xi )∆x ,
n∑i=1
f
(xi−1 + xi
2
)∆x
left-endpoints right-endpoints midpoints
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 6 / 14
![Page 21: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/21.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Approximations
∫ xi
xi−1f (x) dx f (xi−1)∆x f (xi )∆x f
(xi−1+xi
2
)∆x
Rectangle Approximations of∫ xi
xi−1f (x) dx
f (xi−1)∆x , f (xi )∆x , f(
xi−1+xi
2
)∆x
left-endpoints right-endpoints midpoints
Rectangle Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx
n∑i=1
f (xi−1)∆x ,
n∑i=1
f (xi )∆x ,
n∑i=1
f
(xi−1 + xi
2
)∆x
left-endpoints right-endpoints midpoints
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 6 / 14
![Page 22: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/22.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 23: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/23.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 24: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/24.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 25: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/25.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 26: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/26.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 27: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/27.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Approximating ln 2 = 0.69314718 · · ·
L5 = 15
(1 + 5
6 + 57 + 5
8 + 59
)≈ 0.7456
R5 = 15
(56 + 5
7 + 58 + 5
9 + 12
)≈ 0.6456
M5 = 15
(1011 + 10
13 + 1015 + 10
17 + 1019
)≈ 0.6919
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 28: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/28.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Approximating ln 2 = 0.69314718 · · ·
L5 = 15
(1 + 5
6 + 57 + 5
8 + 59
)≈ 0.7456
R5 = 15
(56 + 5
7 + 58 + 5
9 + 12
)≈ 0.6456
M5 = 15
(1011 + 10
13 + 1015 + 10
17 + 1019
)≈ 0.6919
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 29: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/29.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Approximating ln 2 = 0.69314718 · · ·
L5 = 15
(1 + 5
6 + 57 + 5
8 + 59
)≈ 0.7456
R5 = 15
(56 + 5
7 + 58 + 5
9 + 12
)≈ 0.6456
M5 = 15
(1011 + 10
13 + 1015 + 10
17 + 1019
)≈ 0.6919
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 30: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/30.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Rectangle Rules
Rectangle Rule for Approximating∫ ba f (x) dx
Left-endpoint rule:
Ln =b − a
n[f (x0) + f (x1) + · · ·+ f (xn−1)]
Right-endpoint rule:
Rn =b − a
n[f (x1) + f (x2) + · · ·+ f (xn)]
Midpoint rule:
Mn =b − a
n
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Approximating ln 2 = 0.69314718 · · ·
L5 = 15
(1 + 5
6 + 57 + 5
8 + 59
)≈ 0.7456
R5 = 15
(56 + 5
7 + 58 + 5
9 + 12
)≈ 0.6456
M5 = 15
(1011 + 10
13 + 1015 + 10
17 + 1019
)≈ 0.6919
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 7 / 14
![Page 31: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/31.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Error Estimates
∣∣∣∣∫ b
af (x) dx − Ln
∣∣∣∣ ≤ (b − a
n
)[f (b)−f (a)]
Error Estimates
Left-endpoint rule:
ELn =
∫ b
af (x) dx − Ln =
1
2
(b − a)2
nf ′(c) = O(∆x)
Right-endpoint rule:
ERn =
∫ b
af (x) dx − Rn =
1
2
(b − a)2
nf ′(c) = O(∆x)
Midpoint rule:
EMn =
∫ b
af (x) dx −Mn =
1
24
(b − a)3
n2f ′′(c) = O((∆x)2)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 8 / 14
![Page 32: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/32.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Error Estimates
∣∣∣∣∫ b
af (x) dx − Ln
∣∣∣∣ ≤ (b − a
n
)[f (b)−f (a)]
Error Estimates
Left-endpoint rule:
ELn =
∫ b
af (x) dx − Ln =
1
2
(b − a)2
nf ′(c) = O(∆x)
Right-endpoint rule:
ERn =
∫ b
af (x) dx − Rn =
1
2
(b − a)2
nf ′(c) = O(∆x)
Midpoint rule:
EMn =
∫ b
af (x) dx −Mn =
1
24
(b − a)3
n2f ′′(c) = O((∆x)2)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 8 / 14
![Page 33: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/33.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Error Estimates
∣∣∣∣∫ b
af (x) dx − Ln
∣∣∣∣ ≤ (b − a
n
)[f (b)−f (a)]
Error Estimates
Left-endpoint rule:
ELn =
∫ b
af (x) dx − Ln =
1
2
(b − a)2
nf ′(c) = O(∆x)
Right-endpoint rule:
ERn =
∫ b
af (x) dx − Rn =
1
2
(b − a)2
nf ′(c) = O(∆x)
Midpoint rule:
EMn =
∫ b
af (x) dx −Mn =
1
24
(b − a)3
n2f ′′(c) = O((∆x)2)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 8 / 14
![Page 34: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/34.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Error Estimates
∣∣∣∣∫ b
af (x) dx − Ln
∣∣∣∣ ≤ (b − a
n
)[f (b)−f (a)]
Error Estimates
Left-endpoint rule:
ELn =
∫ b
af (x) dx − Ln =
1
2
(b − a)2
nf ′(c) = O(∆x)
Right-endpoint rule:
ERn =
∫ b
af (x) dx − Rn =
1
2
(b − a)2
nf ′(c) = O(∆x)
Midpoint rule:
EMn =
∫ b
af (x) dx −Mn =
1
24
(b − a)3
n2f ′′(c) = O((∆x)2)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 8 / 14
![Page 35: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/35.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Error Estimates: Example
Error Estimates
Left-endpoint rule:
ELn =
∫ b
af (x) dx − Ln =
1
2
(b − a)2
nf ′(c) = O(∆x)
Right-endpoint rule:
ERn =
∫ b
af (x) dx − Rn =
1
2
(b − a)2
nf ′(c) = O(∆x)
Midpoint rule:
EMn =
∫ b
af (x) dx −Mn =
1
24
(b − a)3
n2f ′′(c) = O((∆x)2)
Approximating ln 2 = 0.69314718 · · ·Note that f (x) = 1
x , f ′(x) = − 1x2 , f ′′(x) = 2
x3 .
|ELn | = |ER
n | ≤ 12
(2−1)2
5 maxc∈[1,2] |f ′(c)| = 110
|EMn | ≤ 1
24(2−1)3
52 maxc∈[1,2] |f ′′(c)| = 1300
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 9 / 14
![Page 36: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/36.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Error Estimates: Example
Error Estimates
Left-endpoint rule:
ELn =
∫ b
af (x) dx − Ln =
1
2
(b − a)2
nf ′(c) = O(∆x)
Right-endpoint rule:
ERn =
∫ b
af (x) dx − Rn =
1
2
(b − a)2
nf ′(c) = O(∆x)
Midpoint rule:
EMn =
∫ b
af (x) dx −Mn =
1
24
(b − a)3
n2f ′′(c) = O((∆x)2)
Approximating ln 2 = 0.69314718 · · ·Note that f (x) = 1
x , f ′(x) = − 1x2 , f ′′(x) = 2
x3 .
|ELn | = |ER
n | ≤ 12
(2−1)2
5 maxc∈[1,2] |f ′(c)| = 110
|EMn | ≤ 1
24(2−1)3
52 maxc∈[1,2] |f ′′(c)| = 1300
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 9 / 14
![Page 37: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/37.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsRegular Partition Approximations Error Estimates
Error Estimates: Example
Error Estimates
Left-endpoint rule:
ELn =
∫ b
af (x) dx − Ln =
1
2
(b − a)2
nf ′(c) = O(∆x)
Right-endpoint rule:
ERn =
∫ b
af (x) dx − Rn =
1
2
(b − a)2
nf ′(c) = O(∆x)
Midpoint rule:
EMn =
∫ b
af (x) dx −Mn =
1
24
(b − a)3
n2f ′′(c) = O((∆x)2)
Approximating ln 2 = 0.69314718 · · ·Note that f (x) = 1
x , f ′(x) = − 1x2 , f ′′(x) = 2
x3 .
|ELn | = |ER
n | ≤ 12
(2−1)2
5 maxc∈[1,2] |f ′(c)| = 110
|EMn | ≤ 1
24(2−1)3
52 maxc∈[1,2] |f ′′(c)| = 1300
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 9 / 14
![Page 38: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/38.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Parabolic Approximations
∫ xi
xi−1f (x) dx Trapeziodal Parabolic
Approximations of∫ xi
xi−1f (x) dx
12 [f (xi−1) + f (xi )]∆x , 1
6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]∆x
trapeziodal Parabolic
Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx∑n
i=1∆x2 [f (xi−1) + f (xi )] ,
∑ni=1
∆x6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]Trapeziodal Parabolic
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 10 / 14
![Page 39: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/39.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Parabolic Approximations
∫ xi
xi−1f (x) dx Trapeziodal Parabolic
Approximations of∫ xi
xi−1f (x) dx
12 [f (xi−1) + f (xi )]∆x , 1
6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]∆x
trapeziodal Parabolic
Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx∑n
i=1∆x2 [f (xi−1) + f (xi )] ,
∑ni=1
∆x6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]Trapeziodal Parabolic
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 10 / 14
![Page 40: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/40.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Parabolic Approximations
∫ xi
xi−1f (x) dx Trapeziodal Parabolic
Approximations of∫ xi
xi−1f (x) dx
12 [f (xi−1) + f (xi )]∆x , 1
6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]∆x
trapeziodal Parabolic
Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx∑n
i=1∆x2 [f (xi−1) + f (xi )] ,
∑ni=1
∆x6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]Trapeziodal Parabolic
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 10 / 14
![Page 41: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/41.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Parabolic Approximations
∫ xi
xi−1f (x) dx Trapeziodal Parabolic
Approximations of∫ xi
xi−1f (x) dx
12 [f (xi−1) + f (xi )]∆x , 1
6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]∆x
trapeziodal Parabolic
Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx∑n
i=1∆x2 [f (xi−1) + f (xi )] ,
∑ni=1
∆x6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]Trapeziodal Parabolic
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 10 / 14
![Page 42: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/42.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Parabolic Approximations
∫ xi
xi−1f (x) dx Trapeziodal Parabolic
Approximations of∫ xi
xi−1f (x) dx
12 [f (xi−1) + f (xi )]∆x , 1
6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]∆x
trapeziodal Parabolic
Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx∑n
i=1∆x2 [f (xi−1) + f (xi )] ,
∑ni=1
∆x6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]Trapeziodal Parabolic
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 10 / 14
![Page 43: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/43.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Parabolic Approximations
∫ xi
xi−1f (x) dx Trapeziodal Parabolic
Approximations of∫ xi
xi−1f (x) dx
12 [f (xi−1) + f (xi )]∆x , 1
6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]∆x
trapeziodal Parabolic
Approximations of∫ ba f (x) dx =
∑ni=1
∫ xi
xi−1f (x) dx∑n
i=1∆x2 [f (xi−1) + f (xi )] ,
∑ni=1
∆x6
[f (xi−1) + 4f
(xi−1+xi
2
)+ f (xi )
]Trapeziodal Parabolic
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 10 / 14
![Page 44: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/44.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Simpson’s Rules
Trapeziodal and Simpson’s Rules for Approximating∫ ba f (x) dx
Trapeziodal Rule:
Tn =b − a
2n[f (x0) + 2f (x1) + · · ·+ 2f (xn−1) + f (xn)]
Simpson’s Rule (Parabolic):
Sn =b − a
6n
f (x0) + f (xn) + 2
[f (x1) + · · ·+ 2f (xn−1)
]+4
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 11 / 14
![Page 45: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/45.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Simpson’s Rules
Trapeziodal and Simpson’s Rules for Approximating∫ ba f (x) dx
Trapeziodal Rule:
Tn =b − a
2n[f (x0) + 2f (x1) + · · ·+ 2f (xn−1) + f (xn)]
Simpson’s Rule (Parabolic):
Sn =b − a
6n
f (x0) + f (xn) + 2
[f (x1) + · · ·+ 2f (xn−1)
]+4
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 11 / 14
![Page 46: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/46.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Simpson’s Rules
Trapeziodal and Simpson’s Rules for Approximating∫ ba f (x) dx
Trapeziodal Rule:
Tn =b − a
2n[f (x0) + 2f (x1) + · · ·+ 2f (xn−1) + f (xn)]
Simpson’s Rule (Parabolic):
Sn =b − a
6n
f (x0) + f (xn) + 2
[f (x1) + · · ·+ 2f (xn−1)
]+4
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 11 / 14
![Page 47: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/47.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Simpson’s Rules
Trapeziodal and Simpson’s Rules for Approximating∫ ba f (x) dx
Trapeziodal Rule:
Tn =b − a
2n[f (x0) + 2f (x1) + · · ·+ 2f (xn−1) + f (xn)]
Simpson’s Rule (Parabolic):
Sn =b − a
6n
f (x0) + f (xn) + 2
[f (x1) + · · ·+ 2f (xn−1)
]+4
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Problem
Find the approximate value of ln 2 =∫ 21
dxx
using only the values of f (x) = 1x at
1, 65 , 7
5 , 85 , 9
5 , 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 11 / 14
![Page 48: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/48.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Simpson’s Rules
Trapeziodal and Simpson’s Rules for Approximating∫ ba f (x) dx
Trapeziodal Rule:
Tn =b − a
2n[f (x0) + 2f (x1) + · · ·+ 2f (xn−1) + f (xn)]
Simpson’s Rule (Parabolic):
Sn =b − a
6n
f (x0) + f (xn) + 2
[f (x1) + · · ·+ 2f (xn−1)
]+4
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Approximating ln 2 = 0.69314718 · · ·
T5 = 110
(1 + 10
6 + 107 + 10
8 + 109 + 1
2
)≈ 0.6956
S5 = 130
1 + 1
2 + 2(
56 + 5
7 + 58 + 5
9
)+4
[1011 + 10
13 + 1015 + 10
17 + 1019
]≈ 0.6932
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 11 / 14
![Page 49: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/49.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Simpson’s Rules
Trapeziodal and Simpson’s Rules for Approximating∫ ba f (x) dx
Trapeziodal Rule:
Tn =b − a
2n[f (x0) + 2f (x1) + · · ·+ 2f (xn−1) + f (xn)]
Simpson’s Rule (Parabolic):
Sn =b − a
6n
f (x0) + f (xn) + 2
[f (x1) + · · ·+ 2f (xn−1)
]+4
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Approximating ln 2 = 0.69314718 · · ·
T5 = 110
(1 + 10
6 + 107 + 10
8 + 109 + 1
2
)≈ 0.6956
S5 = 130
1 + 1
2 + 2(
56 + 5
7 + 58 + 5
9
)+4
[1011 + 10
13 + 1015 + 10
17 + 1019
]≈ 0.6932
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 11 / 14
![Page 50: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/50.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Trapeziodal and Simpson’s Rules
Trapeziodal and Simpson’s Rules for Approximating∫ ba f (x) dx
Trapeziodal Rule:
Tn =b − a
2n[f (x0) + 2f (x1) + · · ·+ 2f (xn−1) + f (xn)]
Simpson’s Rule (Parabolic):
Sn =b − a
6n
f (x0) + f (xn) + 2
[f (x1) + · · ·+ 2f (xn−1)
]+4
[f
(x0 + x1
2
)+ · · ·+ f
(xn−1 + xn
2
)]Approximating ln 2 = 0.69314718 · · ·
T5 = 110
(1 + 10
6 + 107 + 10
8 + 109 + 1
2
)≈ 0.6956
S5 = 130
1 + 1
2 + 2(
56 + 5
7 + 58 + 5
9
)+4
[1011 + 10
13 + 1015 + 10
17 + 1019
]≈ 0.6932
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 11 / 14
![Page 51: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/51.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Error Estimates
∣∣∣∣∫ b
af (x) dx − Tn
∣∣∣∣ ≤ 1
2
(b − a
n
)[f (b)−f (a)]
Error Estimates
Trapeziodal Rule:
ETn =
∫ b
af (x) dx − Tn = − 1
12
(b − a)3
n2f ′′(c) = O((∆x)2)
Simpson’s Rule:
ESn =
∫ b
af (x) dx − Sn = − 1
2880
(b − a)5
n4f (4)(c) = O((∆x)4)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 12 / 14
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Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Error Estimates
∣∣∣∣∫ b
af (x) dx − Tn
∣∣∣∣ ≤ 1
2
(b − a
n
)[f (b)−f (a)]
Error Estimates
Trapeziodal Rule:
ETn =
∫ b
af (x) dx − Tn = − 1
12
(b − a)3
n2f ′′(c) = O((∆x)2)
Simpson’s Rule:
ESn =
∫ b
af (x) dx − Sn = − 1
2880
(b − a)5
n4f (4)(c) = O((∆x)4)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 12 / 14
![Page 53: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/53.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Error Estimates
∣∣∣∣∫ b
af (x) dx − Tn
∣∣∣∣ ≤ 1
2
(b − a
n
)[f (b)−f (a)]
Error Estimates
Trapeziodal Rule:
ETn =
∫ b
af (x) dx − Tn = − 1
12
(b − a)3
n2f ′′(c) = O((∆x)2)
Simpson’s Rule:
ESn =
∫ b
af (x) dx − Sn = − 1
2880
(b − a)5
n4f (4)(c) = O((∆x)4)
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 12 / 14
![Page 54: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/54.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Error Estimates: Example
Error Estimates
Trapeziodal Rule:
ETn =
∫ b
af (x) dx − Tn = − 1
12
(b − a)3
n2f ′′(c) = O((∆x)2)
Simpson’s Rule:
ESn =
∫ b
af (x) dx − Sn = − 1
2880
(b − a)5
n4f (4)(c) = O((∆x)4)
Approximating ln 2 = 0.69314718 · · ·Note that f (x) = 1
x , f ′(x) = − 1x2 , f ′′(x) = 2
x3 ,
f (3)(x) = − 6x4 , f (4)(x) = 24
x5 .
|ETn | ≤ 1
12(2−1)3
52 maxc∈[1,2] |f ′′(c)| = 1150
|EMn | ≤ 1
2880(2−1)5
54 maxc∈[1,2] |f (4)(c)| = · · ·
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 13 / 14
![Page 55: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/55.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Error Estimates: Example
Error Estimates
Trapeziodal Rule:
ETn =
∫ b
af (x) dx − Tn = − 1
12
(b − a)3
n2f ′′(c) = O((∆x)2)
Simpson’s Rule:
ESn =
∫ b
af (x) dx − Sn = − 1
2880
(b − a)5
n4f (4)(c) = O((∆x)4)
Approximating ln 2 = 0.69314718 · · ·Note that f (x) = 1
x , f ′(x) = − 1x2 , f ′′(x) = 2
x3 ,
f (3)(x) = − 6x4 , f (4)(x) = 24
x5 .
|ETn | ≤ 1
12(2−1)3
52 maxc∈[1,2] |f ′′(c)| = 1150
|EMn | ≤ 1
2880(2−1)5
54 maxc∈[1,2] |f (4)(c)| = · · ·
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 13 / 14
![Page 56: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/56.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Error Estimates: Example
Error Estimates
Trapeziodal Rule:
ETn =
∫ b
af (x) dx − Tn = − 1
12
(b − a)3
n2f ′′(c) = O((∆x)2)
Simpson’s Rule:
ESn =
∫ b
af (x) dx − Sn = − 1
2880
(b − a)5
n4f (4)(c) = O((∆x)4)
Approximating ln 2 = 0.69314718 · · ·Note that f (x) = 1
x , f ′(x) = − 1x2 , f ′′(x) = 2
x3 ,
f (3)(x) = − 6x4 , f (4)(x) = 24
x5 .
|ETn | ≤ 1
12(2−1)3
52 maxc∈[1,2] |f ′′(c)| = 1150
|EMn | ≤ 1
2880(2−1)5
54 maxc∈[1,2] |f (4)(c)| = · · ·
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 13 / 14
![Page 57: Lecture 11 - Section 8.7 Numerical Integrationjiwenhe/Math1432/lectures/lecture11.pdf · Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008](https://reader033.vdocuments.site/reader033/viewer/2022053023/6056809650dbd533e0211b5f/html5/thumbnails/57.jpg)
Riemann Sums Rectangle Approximations Trapeziodal and Parabolic ApproximationsApproximations Error Estimates
Outline
Riemann SumsArea ProblemLower and Upper SumsRiemann Sum
Rectangle ApproximationsRegular PartitionApproximationsError Estimates
Trapeziodal and Parabolic ApproximationsApproximationsError Estimates
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 11 February 19, 2008 14 / 14