Describe los metodos del trapecio y simpson

1. Dudas de la actividad 3
y evidencia, unidad 2

2. න
−1
4
−𝑥2 + 3𝑥 + 4 𝑑𝑥 =
125
6
= 20.83
Intervalo 𝑋𝑖 ∆𝑋𝑖 𝑓(𝑋𝑖) Área i
i = 0 X0 =- 1 1 -(-1)2+3(-1)+4 = 0 (1)(0) =0
i = 1 X1 = 0 1 -(0)2+3(0)+4 = 4 (1)(4) = 4
i = 2 X2 = 1 1 -(1)2+3(1)+4 = 6 (1)(6) = 6
i = 3 X3 = 2 1 -(2)2+3(2)+4 = 6 (1)(6) = 6
i = 4 X4 = 3 1 -(3)2+3(3)+4 = 4 (1)(4) = 4
Suma 20 u2
∆𝑥 =
4 + 1
5
=
5
5
= 1
𝑥𝑖 = 𝑎 + 𝑖 ∗ ∆𝑥 = −1 + 𝑖

3. න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 =
𝑏 − 𝑎
𝑛
෍
𝑖=1
𝑛 𝑓 𝑥𝑖 + 𝑓 𝑥𝑖 +
𝑏 − 𝑎
𝑛
2
Interv
alo
𝑋𝑖 𝑋𝑖 + ∆𝒙 ∆𝑋𝑖 𝑓(𝑋𝑖) 𝑓(𝑋𝑖 + ∆𝐱) Área i
i = 0 X0
=-
1
X0 = 0 1 -(-1)2+3(-
1)+4 =0
-(0)2+3(0)+4 = 4 ((1)(0) + (1)(4))/2 = 2
i = 1 X1 =
0
X1 = 1 1 -(0)2+3(0)+4
= 4
-(1)2+3(1)+4 = 6 ((1)(4) + (1)(6))/2 =5
i = 2 X2 =
1
X2 = 2 1 -(1)2+3(1)+4
= 6
-(2)2+3(2)+4 = 6 ((1)(6) + (1)(6))/2 = 6
i = 3 X3 =
2
X3 = 3 1 -(2)2+3(2)+4
= 6
-(3)2+3(3)+4 = 4 ((1)(6) + (1)(4))/2 = 5
i = 4 X4 =
3
X4 = 4 1 -(3)2+3(3)+4
= 4
-(4)2+3(4)+4 = 0 ((1)(4) + (1)(0))/2 = 2
Suma 20
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = −𝟏 + 𝒊
∆𝑥 =
4 + 1
5
=
5
5
= 1
න
−1
4
−𝑥2 + 3𝑥 + 4 𝑑𝑥 =
125
6
= 20.83
∆𝒙 =
𝒃 − 𝒂
𝒏

4. 𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟎𝟖𝒊
∆𝑥 =
0.8 − 0
1
= 0.8
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=1
∆𝒙 =
𝒃 − 𝒂
𝒏
i xi f(xi) a 0
0 0 0.2 b 0.8
1 0.8 0.232 n 1
h 0.8
Area 0.1728
Error 1.467733

5.  ‫׬‬−2
4
1 − 𝑥 − 4𝑥3
+ 2𝑥5
𝑑𝑥 =
 ‫׬‬
−2
4
𝑑𝑥 − ‫׬‬−2
4
𝑥𝑑𝑥 − 4 ‫׬‬
−2
4
𝑥3𝑑𝑥 + 2 ‫׬‬
−2
4
𝑥5𝑑𝑥
 𝑥 −
𝑥2
2
−
4𝑥4
4
+
2𝑥6
6
|
4
−2
=> 𝑥 −
𝑥2
2
− 𝑥4
+
1𝑥6
3
|
4
−2
 𝟒 −
𝟒 𝟐
𝟐
− (𝟒)𝟒
+
(𝟒)𝟔
𝟑
− (−𝟐) −
(−𝟐)𝟐
𝟐
− (−𝟐)𝟒
+
(−𝟐)𝟔
𝟑
 𝟒 −
𝟏𝟔
𝟐
− 𝟐𝟓𝟔 +
𝟒𝟎𝟗𝟔
𝟑
− −𝟐 −
𝟒
𝟐
− 𝟏𝟔 +
𝟔𝟒
𝟑
න
𝒂
𝒃
𝒇 𝒙 𝒅𝒙 = 𝑭 𝒃 − 𝑭(𝒂)
𝟏. න 𝒌𝒅𝒙 = 𝒌𝒙 + 𝒄 𝟐. න 𝒄𝒙𝒏
𝒅𝒙 =
𝒄𝒙𝒏+𝟏
𝒏 + 𝟏
, 𝒏 ≠ −𝟏

6. 𝟒 −
𝟏𝟔
𝟐
− 𝟐𝟓𝟔 +
𝟒𝟎𝟗𝟔
𝟑
− −𝟐 −
𝟒
𝟐
− 𝟏𝟔 +
𝟔𝟒
𝟑
𝟒 − 𝟖 − 𝟐𝟓𝟔 +
𝟒𝟎𝟗𝟔
𝟑
+ 𝟐 + 𝟐 + 𝟏𝟔 −
𝟔𝟒
𝟑
−240 +
4096
3
−
64
3
=
−720 + 4096 − 64
3
=
3312
3
= 𝟏𝟏𝟎𝟒
Método analítico

8. 𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟎𝟖𝒊
∆𝑥 =
0.8 − 0
10
= 0.08
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=10
∆𝒙 =
𝒃 − 𝒂
𝒏
i xi F(xi) a 0
0 0 0.2 0 0.8
1 0.08 1.23004672 n 10
2 0.16 1.29691904 h 0.08
3 0.24 1.34372096
4 0.32 1.74339328
5 0.4 2.456
6 0.48 3.18601472
7 0.56 3.53960704
8 0.64 3.18192896
9 0.72 1.99440128
10 0.8 0.232
Área 1.61504256
Error 0.02549044

9. න
𝒂
𝒃
𝒇 𝒙 𝒅𝒙 =
𝒃 − 𝒂
𝒏
෍
𝒊=𝟏
𝒏 𝒇 𝒙𝒊 + 𝒇 𝒙𝒊 +
𝒃 − 𝒂
𝒏
𝟐
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟎𝟖𝒊
∆𝑥 =
0.8 − 0
10
= 0.08
‫׬‬
0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=10
∆𝒙 =
𝒃 − 𝒂
𝒏
i xi xi+ ∆x ∆xi F(xi) f(Xi+x) Area
0 0 0.08 0.08 0.2 1.23004672 0.05720187

10. න
𝒂
𝒃
𝒇 𝒙 𝒅𝒙 =
𝒃 − 𝒂
𝒏
෍
𝒊=𝟏
𝒏 𝒇 𝒙𝒊 + 𝒇 𝒙𝒊 +
𝒃 − 𝒂
𝒏
𝟐 𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟎𝟖𝒊
∆𝑥 =
0.8 − 0
10
= 0.08
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=10
∆𝒙 =
𝒃 − 𝒂
𝒏
i xi xi+ ∆x ∆x F(xi) f(Xi+x) Area
0 0 0.08 0.08 0.2 1.23004672 0.05720187
1 0.08 0.16 0.08 1.23004672 1.29691904 0.10107863
2 0.16 0.24 0.08 1.29691904 1.34372096 0.1056256
3 0.24 0.32 0.08 1.34372096 1.74339328 0.12348457
4 0.32 0.4 0.08 1.74339328 2.456 0.16797573
5 0.4 0.48 0.08 2.456 3.18601472 0.22568059
6 0.48 0.56 0.08 3.18601472 3.53960704 0.26902487
7 0.56 0.64 0.08 3.53960704 3.18192896 0.26886144
8 0.64 0.72 0.08 3.18192896 1.99440128 0.20705321
9 0.72 0.8 0.08 1.99440128 0.232 0.08905605
Total Área 1.61504256

11. න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑏 − 𝑎
𝑓(𝑥0) + 𝟒𝒇(𝒙𝟏) + 𝒇(𝒙𝟐)
6
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟖𝒊
∆𝑥 =
0.8 − 0
2
= 0.4
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=2 ∆𝒙 =
𝒃 − 𝒂
𝒏
i xi f(xi) a 0
0 0 0.2 0.2 b 0.8
1 0.4 2.456 9.824 n 2
2 0.8 0.232 0.232 h 0.4
10.256
Exacto 1.640533
Integral 1.36746667
Error 0.27306633
Error total
-
0.12014933

12. න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑏 − 𝑎
𝑓(𝑥0) + 𝟒 σ𝒊=𝟏,𝟑,𝟓
𝒏−𝟏
𝒇(𝒙𝒊) + 𝟐 σ𝒊=𝟐,𝟒,𝟔
𝒏−𝟐
𝒇(𝒙𝒊) + 𝒇(𝒙𝒏)
3𝑛
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟖𝒊
∆𝑥 =
0.8 − 0
4
= 0.2
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=4 ∆𝒙 =
𝒃 − 𝒂
𝒏
i xi f(xi) a 0
0 0 0.2 0.2 b 0.8
1 0.2 1.288 19.008 n 4
2 0.4 2.456 4.912 h 0.2
3 0.6 3.464 0.232
4 0.8 0.232
Integral 1.62346667 0.01706633

13. න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑏 − 𝑎
𝑓(𝑥0) + 𝟒 σ𝒊=𝟏,𝟑,𝟓
𝒏−𝟏
𝒇(𝒙𝒊) + 𝟐 σ𝒊=𝟐,𝟒,𝟔
𝒏−𝟐
𝒇(𝒙𝒊) + 𝒇(𝒙𝒏)
3𝑛
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟖𝒊
∆𝑥 =
0.8 − 0
10
= 0.8
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=10 ∆𝒙 =
𝒃 − 𝒂
𝒏
i xi f(xi) a 0
0 0 0.2 0.2 b 0.8
1 0.08 1.23004672 n 10
2 0.16 1.29691904 h (∆𝒙) 0.08
3 0.24 1.34372096 4(10.563776)=42.255104
4 0.32 1.74339328 2(9.408256)=18.816512
5 0.4 2.456
6 0.48 3.18601472 61.503616
7 0.56 3.53960704 2.050120533
8 0.64 3.18192896 1.640096427 1.64009643 0.00043657
9 0.72 1.99440128 1.64009643
10 0.8 0.232

14. න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 =
∆𝒙
3
[𝑓(𝑥0) + 𝟒 ෍
𝒊=𝟏,𝟑,𝟓
𝒏−𝟏
𝒇(𝒙𝒊) + 𝟐 ෍
𝒊=𝟐,𝟒,𝟔
𝒏−𝟐
𝒇(𝒙𝒊) + 𝒇(𝒙𝒏)]
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟖𝒊
∆𝑥 =
0.8 − 0
10
= 0.8
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=10 ∆𝒙 =
𝒃 − 𝒂
𝒏
i xi f(xi) a 0
0 0 0.2 0.2 b 0.8
1 0.08 1.23004672 n 10
2 0.16 1.29691904 h (∆𝒙) 0.08
3 0.24 1.34372096 4(10.563776)=42.255104
4 0.32 1.74339328 2(9.408256)=18.816512
5 0.4 2.456
6 0.48 3.18601472 61.503616
7 0.56 3.53960704
8 0.64 3.18192896 1.640096427 1.64009643 0.00043657
9 0.72 1.99440128 1.64009643
10 0.8 0.232

15. න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 =
𝑏 − 𝑎
𝑛
[𝑓(𝑥0) + 4 ෍
𝑖=1,3,5
𝑛−1
𝑓(𝑥𝑖) + 2 ෍
𝑖=2,4,6
𝑛−2
𝑓(𝑥𝑖) + 𝑓(𝑥𝑛)]
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟖𝒊
∆𝑥 =
0.8 − 0
10
= 0.8
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=10 ∆𝒙 =
𝒃 − 𝒂
𝒏
i xi f(xi) a 0
0 0 0.2 b 0.8
1 0.08 1.23004672 n 10
2 0.16 1.29691904 h 0.08
3 0.24 1.34372096
4 0.32 1.74339328
5 0.4 2.456
6 0.48 3.18601472
7 0.56 3.53960704
8 0.64 3.18192896 Integral 1.64009643 0.00043657
9 0.72 1.99440128 1.64009643
10 0.8 0.232

16. න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑏 − 𝑎
𝑓(𝑥0) + 3𝑓(𝑥1) + 3𝑓(𝑥2) + 𝒇(𝒙𝟑)
8
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟖𝒊
∆𝑥 =
0.8 − 0
5
= 0.16
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=5 ∆𝒙 =
𝒃 − 𝒂
𝒏
i xi f(xi) a 0
0 0 0.2 0.2 b 0.8
1 0.16 1.29691904 17.931735 n 5
2 0.32 1.74339328 3.48678656 h 0.16
3 0.48 3.18601472 3.18192896
4 0.64 3.18192896
5 0.8 0.232
Integral 1/3 0.3803237
Integral 3/8 1.26475346 -0.00454416
Total 1.64507716
න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑏 − 𝑎
𝑓(𝑥0) + 𝟒𝒇(𝒙𝟏) + 𝒇(𝒙𝟐)
6

17. න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑏 − 𝑎
𝑓(𝑥0) + 𝟒 σ𝒊=𝟏,𝟑,𝟓
𝒏−𝟏
𝒇(𝒙𝒊) + 𝟐 σ𝒊=𝟐,𝟒,𝟔
𝒏−𝟐
𝒇(𝒙𝒊) + 𝒇(𝒙𝒏)
3𝑛
𝒙𝒊 = 𝒂 + 𝒊 ∗ ∆𝒙 = 𝟎 + 𝟎. 𝟖𝒊
∆𝑥 =
0.8 − 0
10
= 0.8
‫׬‬0
0.8
(0.2 + 25𝑥 − 200𝑥2
+ 675𝑥3
− 900𝑥4
+ 400𝑥5
)𝑑𝑥 = 1.640533, n=10 ∆𝒙 =
𝒃 − 𝒂
𝒏
i xi f(xi) a 0
0 0 0.2 0.2 b 0.8
1 0.08 1.23004672 n 10
2 0.16 1.29691904 h (∆𝒙) 0.08
3 0.24 1.34372096 4(10.563776)=42.255104
4 0.32 1.74339328 2(9.408256)=18.816512
5 0.4 2.456
6 0.48 3.18601472 61.503616
7 0.56 3.53960704 2.050120533
8 0.64 3.18192896 1.640096427 1.64009643 0.00043657
9 0.72 1.99440128 1.64009643
10 0.8 0.232

18. https://www.youtube.com/watch?v=4pCS8pLgqwI

19. Método de Simpson 1/3
Método de integración para calcular integrales definidas donde se conectan
grupos sucesivos de tres puntos sobre la curva mediante parábolas de segundo
grado. A las fórmulas que resultan de calcular la integral bajo estos polinomios
se les llama Reglas de Simpson.