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 $1-8 .$ ) $$\int_{-1}^{1}\left(x^{2}+1\right) d x$$

Key Concepts

Composite Simpson’s Rule

Simpson’s Rule enhances the accuracy of numerical integration by approximating the function with quadratic polynomials over pairs of subintervals rather than straight lines. Its error analysis involves the fourth derivative of the function, yielding an error that decreases with the fourth power of the subinterval width. This greater sensitivity to changes in the function’s curvature typically makes Simpson's Rule more accurate than the Trapezoidal Rule for smooth functions when using an appropriate number of subintervals.

Error Bound Formulas for Numerical Integration

Error bound formulas are crucial in choosing the number of subintervals to ensure that the calculated integral approximation meets a predefined error tolerance. For the composite Trapezoidal Rule, the error bound involves the maximum second derivative of the function, while for Simpson's Rule, it involves the maximum fourth derivative. These formulas express the error as a function of the interval’s length, the number of subintervals, and the relevant derivative bounds, allowing one to set up and solve inequalities that provide a minimum subinterval count needed to achieve the desired accuracy.

Composite Trapezoidal Rule

This numerical integration method approximates a definite integral by dividing the interval into equal subintervals and approximating the area under the curve as a series of trapezoids. The key aspect of this rule in error analysis is that its error is proportional to the square of the subinterval width and depends on the maximum absolute value of the second derivative of the function over the interval. This property allows one to determine how many subintervals are needed to achieve a specified error tolerance.

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