66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
70. Let f(x) = e^(-x²).
a. Find a Simpson's Rule approximation to the integral from 0 to 3 of e^(-x²) dx using n = 30 subintervals.

Verified step by step guidance

1. Step 1: Recall Simpson's Rule formula for approximating integrals: \( \int_{a}^{b} f(x) dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n) \right] \), where \( h = \frac{b-a}{n} \) and \( x_i = a + i \cdot h \).
2. Step 2: Identify the parameters for the problem. Here, \( f(x) = e^{-x^2} \), \( a = 0 \), \( b = 3 \), and \( n = 30 \). Calculate \( h \) using \( h = \frac{b-a}{n} \), which gives \( h = \frac{3-0}{30} = 0.1 \).
3. Step 3: Generate the \( x_i \) values. These are the points \( x_0, x_1, x_2, \dots, x_{30} \), where \( x_i = a + i \cdot h \). For example, \( x_0 = 0 \), \( x_1 = 0.1 \), \( x_2 = 0.2 \), and so on up to \( x_{30} = 3 \).
4. Step 4: Evaluate \( f(x_i) \) for each \( x_i \). Compute \( f(x_i) = e^{-x_i^2} \) for all \( x_i \) values. For example, \( f(x_0) = e^{-(0)^2} = 1 \), \( f(x_1) = e^{-(0.1)^2} \), and so on.
5. Step 5: Apply Simpson's Rule formula. Substitute the \( f(x_i) \) values into the formula, ensuring the coefficients alternate between 4 and 2 for interior points, with \( f(x_0) \) and \( f(x_{30}) \) having coefficients of 1. Multiply by \( \frac{h}{3} \) to complete the approximation.

Key Concepts

Simpson's Rule

Simpson's Rule is a numerical method for approximating the definite integral of a function. It uses parabolic segments to estimate the area under the curve, providing a more accurate approximation than methods like the Trapezoidal Rule. The formula involves evaluating the function at evenly spaced points and applying weights to these values, making it particularly effective for smooth functions.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the total accumulation of the function's values across the interval.

Error Estimation

Error estimation in numerical integration involves determining how closely the approximation of an integral matches the actual value. Theorem 8.1 likely provides a framework for estimating the error associated with Simpson's Rule, which can depend on factors such as the number of subintervals and the behavior of the function being integrated. Understanding error helps in assessing the reliability of the numerical approximation.