In Exercises 11-22, estimate the minimum number of...

In Exercises $11-22,$ estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than $10^{-4}$ by (a) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises $11-18$ are the integrals from Exercises $1-8 .$ ) $$\int_{1}^{3}(2 x-1) d x$$

Key Concepts

Simpson's Rule Error Bound

This concept relates to estimating the error in numerical integration when using Simpson's rule. The error bound for Simpson’s rule is given by |E_S| ? K*(b ? a)?/(180*n?), where K is the maximum absolute value of the fourth derivative of the function on the interval. This error estimate allows one to calculate the minimum number of subintervals required for the approximation to meet the desired precision.

Determining the Number of Subintervals for Desired Accuracy

This concept involves using the error bounds of numerical integration methods to choose the appropriate number of subintervals so that the absolute error remains below a predetermined tolerance (e.g., 10??). By determining an upper bound for the relevant derivative (second for the trapezoidal rule and fourth for Simpson's rule) on the interval, one can solve the corresponding inequality to find the minimum n, ensuring computational efficiency and accuracy in the approximation.

Trapezoidal Rule Error Bound

This concept involves estimating the error when approximating an integral using the trapezoidal rule. The error bound formula for the trapezoidal rule, |E_T| ? K*(b ? a)³/(12*n²), uses the maximum absolute value of the second derivative of the function over the interval [a, b]. It helps determine how many subintervals are needed to ensure that the approximation error is below a specified tolerance.

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