Question

Estimate the minimum number of subintervals to approximate the value of Integral from negative 3 to 3 (2 x squared plus 9 )dx with an error of magnitude less than 4 times 10 Superscript negative 4 using a. the error estimate formula for the Trapezoidal Rule. b. the error estimate formula for​ Simpson's Rule.

Answer 1

Answer:

The value of n for (a) n=425. (b) n=0.

Step-by-step explanation:

Given integration is,

$\int_{-3}^{3}f(x)=\int_{-3}^{3}(2x^2+9)dx$

(a) For Trapezoidal Rule : Composite error is,

$E_T=-\frac{(b-a)^3}{12n^2}f''(x_m)$

Where, f''(x_m)=greatest value of |f''(x)|= |4|=4, a=-3, b=3. Therefore to find the minimum number of subinterval,

$|E_T|\leq | \frac{(b-a)^3}{12n^2}f''(x)|$|

$\leq \frac{4(-3-3)^3}{12n^2}$

$=\frac{72}{n^2}$

According to the question, we must choose n such that,

$\frac{72}{n^2}$

$\implies n>424.2640686$

So we can take n=425.

(b) For Simpson 1/3 rule : Composite error is,

$E_S=-\frac{(b-a)^5}{180n^4}f^{(iv)}(x_m)$

where, $f^{(x_m)}=\textit{greatest value of} |f^{(iv)}(x)|$

In this problem $f^{(iv)}(x)=0$, so that,

$|E_S|\leq 0$

that is there exist no error. So n=0.