Answer:
The value of n for (a) n=425. (b) n=0.
Step-by-step explanation:
Given integration is,
![\int_{-3}^{3}f(x)=\int_{-3}^{3}(2x^2+9)dx](https://tex.z-dn.net/?f=%5Cint_%7B-3%7D%5E%7B3%7Df%28x%29%3D%5Cint_%7B-3%7D%5E%7B3%7D%282x%5E2%2B9%29dx)
(a) For Trapezoidal Rule : Composite error is,
![E_T=-\frac{(b-a)^3}{12n^2}f''(x_m)](https://tex.z-dn.net/?f=E_T%3D-%5Cfrac%7B%28b-a%29%5E3%7D%7B12n%5E2%7Df%27%27%28x_m%29)
Where, f''(x_m)=greatest value of |f''(x)|= |4|=4, a=-3, b=3. Therefore to find the minimum number of subinterval,
|
![\leq \frac{4(-3-3)^3}{12n^2}](https://tex.z-dn.net/?f=%5Cleq%20%5Cfrac%7B4%28-3-3%29%5E3%7D%7B12n%5E2%7D)
![=\frac{72}{n^2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B72%7D%7Bn%5E2%7D)
According to the question, we must choose n such that,
![\frac{72}{n^2}](https://tex.z-dn.net/?f=%5Cfrac%7B72%7D%7Bn%5E2%7D%3C4%5Ctimes%2010%5E%7B-4%7D)
![\implies n>424.2640686](https://tex.z-dn.net/?f=%5Cimplies%20n%3E424.2640686)
So we can take n=425.
(b) For Simpson 1/3 rule : Composite error is,
![E_S=-\frac{(b-a)^5}{180n^4}f^{(iv)}(x_m)](https://tex.z-dn.net/?f=E_S%3D-%5Cfrac%7B%28b-a%29%5E5%7D%7B180n%5E4%7Df%5E%7B%28iv%29%7D%28x_m%29)
where,
In this problem
, so that,
![|E_S|\leq 0](https://tex.z-dn.net/?f=%7CE_S%7C%5Cleq%200)
that is there exist no error. So n=0.