a) Trapezoidal rule

b) Simpsons 1/3rd rule

c) Simpsons 3/8th rule

Mumbai University > First Year Engineering > sem 2 > Applied Maths 2

Marks : 8

Year : DEC 2015

Let $a=0 ,b=6$ and $n=6\\ \therefore h=\dfrac {b-a}n=\dfrac {6-0}6=1\\ Let \space y=e^x$

1) Trapezoidal rule

$\int\limits_a^b f(x)dx=\dfrac h2[(y_0+y_6)+2(y_1+y_2+y_3+y_4+y_5)]\\ =\dfrac 12[(1+403.4)+2(2.718+7.38+20.08+54.59+148.4)]\\ =\dfrac 12 [404.4+2(233.168)]\\ =435.368$

2) Simpsons 1/3rd rule

$\int\limits_a^b f(x)dx=\dfrac h3[(y_0+y_6)+2(y_1+y_3+y_5)+2(y_2+y_4)]\\ =\dfrac 13[(1+403.4)+4(2.718+20.08+148.4)+2(54.59+7.38)]\\ =\dfrac 13 [404.4+4\times 171.198 +2(62.28)]\\ =404.584$

3) Simpsons 3/8th rule

$\int\limits_a^b f(x)dx=\dfrac {3h}8[(y_0+y_6)+2(y_1+y_4+y_5)+2(y_3)]\\ =\dfrac 38[(1+403.4)+3(2.718+54.59+7.38+148.4)+2(20.08)]\\ =\dfrac [404.4+3(213.088) + 40.16]\\ =406.434$