Question

Find the minimum number of intervals required to evaluate int (e^(-x^2 +1))dx with an accuracy of 0.5 × 10^(-4) , by using Trapezoidal rule.

Answer 1

To find the minimum number $n$ of intervals, we need to solve the inequality

$$E\leq\frac{(b-a)^3}{12n^2}[\text{max}|f"(x)|],a\leq x\leq b.$$

Therefore, first find the second derivative:

$$\frac{d}{dx}\big[e^{-x^2+1}\big]=-2xe^{1-x^2}=f'(x),\\ \space\\ \frac{d}{dx}\big[-2xe^{1-x^2}\big]=2e^{1-x^2}(2x^2-1)=f"(x).$$

Assume that the integration limits are $a=-1, b=2$ because they are not present in the condition. Therefore:

$f"(-1)=2\\ f"(2)=0.7\\ f"(0)=-5.43656...=\text{max}[f"(x)].\\$

Find $n$:

$$0.5\cdot10^{-4}\leq\frac{(2-(-1))^3}{12n^2}[5.27]=\frac{27}{12n^2}\cdot5.4366, \\ \space\\ n^2\geq244647,\\n\geq494.62\approx496.$$