Trapezoid Rule and Simpson's Rule Consider the following...

Trapezoid Rule and Simpson's Rule Consider the following integrals and the given values of $n .$ a. Find the Trapezoid Rule approximations to the integral using $n$ and $2 n$ subintervals. b. Find the Simpson's Rule approximation to the integral using $2 n$ subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example $8 .$ c. Compute the absolute errors in the Trapezoid Rule and Simpson's Rule with $2 n$ subintervals. $$\int_{0}^{1} e^{2 x} d x, n=25$$

Trapezoid Rule and Simpson's Rule Consider the following integrals and the given values of $n .$
a. Find the Trapezoid Rule approximations to the integral using $n$ and $2 n$ subintervals.
b. Find the Simpson's Rule approximation to the integral using $2 n$ subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example $8 .$
c. Compute the absolute errors in the Trapezoid Rule and Simpson's Rule with $2 n$ subintervals.
$$\int_{0}^{1} e^{2 x} d x, n=25$$

Key Concepts

Numerical Integration

Numerical integration is the process of approximating the value of a definite integral, especially when an antiderivative is difficult or impossible to obtain analytically. It involves partitioning the interval of integration into smaller subintervals and then summing the contributions from each subinterval using an appropriate rule or formula.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method for estimating the value of a definite integral by approximating the area under the curve as a series of trapezoids. The method works by dividing the interval into smaller subintervals, computing the function’s value at the endpoints, and then calculating the area of each trapezoid. The approximation improves as the number of subintervals increases.

Simpson's Rule

Simpson's Rule is an enhanced numerical integration technique that approximates the integrals by fitting parabolic arcs over pairs of subintervals rather than straight lines. It is generally more accurate than the Trapezoidal Rule for smooth functions, as it takes into account the curvature of the function by using quadratic interpolation. Simpson's Rule usually requires an even number of subintervals.

Error Analysis

Error analysis in numerical integration involves estimating the difference between the true value of the integral and its numerical approximation. This often includes computing absolute errors, which quantify the magnitude of the discrepancy regardless of its sign, and helps in assessing the reliability and accuracy of the chosen numerical method.

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