Question

find the minimum number of intervals required to evaluate integration 0 to 1 e^-xdx with an accuracy of 1/2 X 10^-4, by using the trapezoidal rule.

Answer 1

Question #75299, Math / Other

find the minimum number of intervals required to evaluate integration 0 to $1\mathrm{e}^{\wedge} - \mathrm{x}^{\wedge}2$ dx with an accuracy of $1 / 2 \times 10^{\wedge} - 4$ by using trapezoidal rule

Answer.

$$| \varepsilon | \leq \frac {(b - a) ^ {3}}{1 2 N ^ {2}} \max _ {a \leq x \leq b} f ^ {\prime \prime} (x).$$ $$a = 0, b = 1.$$ $$f (x) = e ^ {- x ^ {2}}, f ^ {\prime} (x) = - 2 x e ^ {- x ^ {2}}, f ^ {\prime \prime} (x) = (4 x ^ {2} - 2) e ^ {- x ^ {2}}.$$ $$\max _ {0 \leq x \leq 1} f ^ {\prime \prime} (x) = f ^ {\prime \prime} (1) = \frac {2}{e} \approx 0. 7 3 5 8.$$ $$\text {S o}, 0. 5 \times 1 0 ^ {- 4} \leq \frac {0 . 7 3 5 8}{1 2 N ^ {2}} \rightarrow N \geq \sqrt {\frac {0 . 7 3 5 8}{1 2 * 0 . 5 \times 1 0 ^ {- 4}}} \approx 3 5.$$

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