Answer to Question #108305 in Quantitative Methods for Garima Ahlawat

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

Find $n$:

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