Answer:
The value of n for (a) n=425. (b) n=0.
Step-by-step explanation:
Given integration is,
[tex]\int_{-3}^{3}f(x)=\int_{-3}^{3}(2x^2+9)dx[/tex]
(a) For Trapezoidal Rule : Composite error is,
[tex]E_T=-\frac{(b-a)^3}{12n^2}f''(x_m)[/tex]
Where, f''(x_m)=greatest value of |f''(x)|= |4|=4, a=-3, b=3. Therefore to find the minimum number of subinterval,
[tex]|E_T|\leq | \frac{(b-a)^3}{12n^2}f''(x)|[/tex]|
[tex]\leq \frac{4(-3-3)^3}{12n^2}[/tex]
[tex]=\frac{72}{n^2}[/tex]
According to the question, we must choose n such that,
[tex]\frac{72}{n^2}<4\times 10^{-4}[/tex]
[tex]\implies n>424.2640686[/tex]
So we can take n=425.
(b) For Simpson 1/3 rule : Composite error is,
[tex]E_S=-\frac{(b-a)^5}{180n^4}f^{(iv)}(x_m)[/tex]
where, [tex]f^{(x_m)}=\textit{greatest value of} |f^{(iv)}(x)|[/tex]
In this problem [tex]f^{(iv)}(x)=0[/tex], so that,
[tex]|E_S|\leq 0[/tex]
that is there exist no error. So n=0.
