Question

Using n=6 integrate the function in the interval [1, 2] 

(i) Complete the table

 x        1       
f(x) 

(ii) Use trapezoidal rule.

(iii) Simpsons rule to evaluate the integral.

Answer 1

Using n=6 integrate the function in the interval [1, 2] 

(i) Complete the table

$\begin{matrix} x & 1 & \frac{7}{6} & \frac{8}{6} & \frac{9}{6} & \frac{10}{6} & \frac{11}{6} & 2\\ f(x) & e^{-\frac{a}{100}}cos\ 3a & e^{-\frac{7a}{600}}cos\ \frac{21a}{6} & e^{-\frac{a}{75}}cos\ 4a & e^{-\frac{3a}{200}}cos\ \frac{9a}{2} & e^{-\frac{a}{60}}cos\ 5a & e^{-\frac{11a}{600}}cos\ \frac{11a}{2} & e^{-\frac{a}{50}}cos\ 6a \end{matrix}$

(ii) Use trapezoidal rule.

$\intop_1^2 e^\frac{-ax}{100}\ cos\ 3ax\ dx=\frac{h}{2}(y_0+2(y_1+y_2+y_3+y_4+y_5)+y_6)\\ =\frac{1}{12}(e^\frac{-a}{100}cos3a+2(e^\frac{-7a}{600}cos\ \frac{27a}{6}+e ^\frac{-a}{75}cos\ 4a+ e^\frac{-3a}{300}cos \frac{9a}{2}+e ^\frac{-a}{60}cos\ 5a+ e^\frac{-11a}{600}cos \frac{11a}{2})+e^\frac{-a}{50}cos 6a)$

(iii) Simpsons rule to evaluate the integral.

$h=\frac{2-1}{6}=\frac{1}{6}$

$\intop_1^2 e^\frac{-ax}{100}\ cos\ 3ax\ dx=\frac{h}{3}(y_0+4(y_1+y_3+y_5)+2(y_2+y_4)+y_6)\\ =\frac{1}{18}(e^\frac{-a}{100}cos3a+4(e^\frac{-7a}{600}cos\ \frac{27a}{6}+e ^\frac{-a}{60}cos\ 5a+ e^\frac{-11a}{600}cos \frac{11a}{2})+2(e ^\frac{-a}{75}cos\ 4a+ e ^\frac{-a}{60}cos\ 5a )+e^\frac{-a}{50}cos 6a)$