This document demonstrates using Simpson's 3/8 rule to approximate two integrals. For the integral from 0 to 1 of (1+x), Simpson's 3/8 rule gives a result of 0.69320, which approximates the exact solution of 0.69315. For the integral from 4 to 5.2 of ln(x), Simpson's 3/8 rule gives a result of 1.82784.

1. 1

2. Topic
Simpson’s 3/8
Method

3. Presented By
Anika Ohab
ID:142-15-3568
Abul Hasnath
ID:142-15-3532
Umme Habiba
ID: 142-15-3677
Shahinur Rahman
ID: 142-15-3606

4. 𝟎
𝟏 𝟏
(𝟏+𝒙)
𝒅𝒙 𝒖𝒔𝒊𝒏𝒈 𝒉 =
𝟏
𝟔
𝑪𝒐𝒓𝒓𝒆𝒄𝒕 𝒖𝒑 𝒕𝒐 𝟒 𝒅𝒆𝒄𝒊𝒎𝒂𝒍𝒆 𝒑𝒍𝒂𝒄𝒆𝒔 𝒃𝒚 𝒖𝒔𝒊𝒏𝒈 𝒔𝒊𝒎𝒑𝒐𝒏′𝒔
𝟑
𝟖
𝑹𝒖𝒍𝒆.
Solve:
𝑿 𝒏 = 𝑿 𝟎 + 𝒏𝒉
1=0+n*
𝟏
𝟔
n= 6
X y=
1
(1+𝑥)
0 1
0.16667 0.85714
0.33334 0.74999
0.50001 0.66666
0.66668 0.59999
0.83335 0.54545
0.1 0.5

5. Simpsons 3/8 rule:
I=
𝟑𝒉
𝟖
[𝒚 𝟎+𝟑𝒚 𝟏+3𝒚 𝟐+2𝒚 𝟑 + 𝟑𝒚 𝟒 + 𝟑𝒚 𝟓+𝒚 𝟔]
=0.69320
Exact ∶ 𝟎
𝟏 𝟏
(𝟏+𝒙)
𝒅𝒙
= [𝑙𝑛(1+x) ] 𝟎
𝟏
= 𝑙𝑛2- 𝑙𝑛1
=.69315

6. 4
5.2
𝑙𝑛 𝑑𝑥 𝑢𝑠𝑖𝑛𝑔 𝑛 = 6 𝐶𝑜𝑟𝑟𝑒𝑐𝑡 𝑢𝑝 𝑡𝑜 4 𝑑𝑒𝑐𝑖𝑚𝑎𝑙𝑒 𝑝𝑙𝑎𝑐𝑒𝑠 𝑏𝑦 𝑢𝑠𝑖𝑛𝑔 𝑠𝑖𝑚𝑝𝑜𝑛𝑠
3
8
𝑅𝑢𝑙𝑒
Solve:
𝑋 𝑛 = 𝑋0 + 𝑛ℎ
5.2=4+6xh
h= .2
X y=lnx
4 1.38629
4.2 1.43508
4.4 1.48160
4.6 1.52605
4.8 1.56862
5.0 1.60943
5.2 1.64865

7. Simpsons 3/8 rule:
I=3h/8[𝑦0+3𝑦1+3𝑦2+2𝑦3 + 3𝑦4 + 3𝑦5+𝑦6]
=1.82784