You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.
You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?
(Remember that for Simpson’s Rule the number of subintervals must be even.)

검증된 단계별 안내

Recall the error bound formula for Simpson's Rule: the error \( E_S \) satisfies \( |E_S| \leq \frac{K(b - a)^5}{180 n^4} \), where \( K \) is an upper bound on \( |f^{(4)}(x)| \) over \([a, b]\), and \( n \) is the number of subintervals (which must be even).

Identify the given values: \( a = 1 \), \( b = 2 \), \( K = 3 \), and the desired error bound \( |E_S| < 10^{-5} \).

Set up the inequality for the error bound: \( \frac{3 (2 - 1)^5}{180 n^4} < 10^{-5} \).

Simplify the inequality to solve for \( n^4 \): \( \frac{3}{180 n^4} < 10^{-5} \) which leads to \( n^4 > \frac{3}{180 \times 10^{-5}} \).

Calculate the right side and then take the fourth root to find \( n \). Finally, choose the smallest even integer \( n \) that satisfies this inequality to ensure the error is less than \( 10^{-5} \).

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주요 개념

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Simpson’s Rule

Simpson’s Rule is a numerical method for approximating definite integrals by fitting parabolas through subintervals of the function. It requires an even number of subintervals and generally provides more accurate results than the trapezoidal rule for smooth functions.

Error Bound for Simpson’s Rule

The error bound for Simpson’s Rule depends on the fourth derivative of the function. Specifically, the error magnitude is bounded by (K(b−a)^5) / (180 n^4), where K is the maximum absolute value of the fourth derivative on [a, b], and n is the number of subintervals.

Choosing Number of Subintervals for Desired Accuracy

To ensure the approximation error is below a specified tolerance, solve the error bound inequality for n, the number of subintervals. Since Simpson’s Rule requires n to be even, round up to the nearest even integer after calculating the minimum n needed.