Find a bound for the error in the approximation S10 to ∫0^3 e^-x^2 d x (use Figure 18 to determi

step 1 Based on the figure, K4 is bounded above by 2. That means that the error in S -10, Simpson's rul

Find a bound for the error in the approximation S10 to ∫0^3...

Find a bound for the error in the approximation $S_{10}$ to $\int_{0}^{3} e^{-x^{2}} d x$ (use Figure 18 to determine a value of $K_{4}$ . Then find a value of $N$ such that $S_{N}$ has an error of at most $10^{-6}$ .

Key Concepts

Simpson's Rule

Simpson's rule is a numerical integration method that approximates the value of a definite integral by using quadratic polynomials to approximate the integrand over subintervals. This method is particularly popular because it generally provides a high degree of accuracy with a relatively small number of subintervals when the integrand is sufficiently smooth.

Error Bound Formula for Simpson's Rule

The error in Simpson's Rule can be estimated by an error bound formula that involves the fourth derivative of the function being integrated. Specifically, the bound is given by (K?(b - a)?)/(180N?), where K? is an upper bound on the absolute value of the fourth derivative over the interval, (b - a) is the length of the interval, and N is the (even) number of subintervals used. This formula is central to understanding and controlling the accuracy of the approximation.

Estimation of the Fourth Derivative

To apply the error bound formula, it is necessary to compute or estimate an upper limit on the absolute value of the fourth derivative of the integrand, denoted by K?, over the interval of integration. This step usually involves finding expressions for higher-order derivatives and then determining their maximum absolute value, which can sometimes be assisted by graphical or analytical tools. This concept is crucial in ensuring that the error bound is both reliable and tight.

Determining the Required Number of Subintervals

Given a desired tolerance for the error in the approximation, the error bound formula can be rearranged to solve for N, the number of subintervals required to achieve the specified accuracy. This involves algebraic manipulation and often solving an inequality to find the minimum even integer N that ensures the error does not exceed the targeted threshold. This approach is a key aspect of ensuring efficient and precise numerical integration.

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