66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
69. Let f(x) = sin(eˣ).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.

Verified step by step guidance

1. Identify the integral to approximate: \(\int_0^1 \sin(e^x) \, dx\) and note that we will use the Trapezoid Rule with \(n = 40\) subintervals.
2. Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n} = \frac{1 - 0}{40} = \frac{1}{40}\).
3. Determine the \(x\)-values at which to evaluate the function: \(x_i = a + i \Delta x = 0 + i \times \frac{1}{40}\) for \(i = 0, 1, 2, \ldots, 40\).
4. Evaluate the function \(f(x) = \sin(e^x)\) at each \(x_i\) to get \(f(x_i) = \sin(e^{x_i})\) for all \(i\) from 0 to 40.
5. Apply the Trapezoid Rule formula: \(T_n = \frac{\Delta x}{2} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]\), substituting the values of \(f(x_i)\) and \(\Delta x\) to set up the approximation.

Key Concepts

Trapezoid Rule

The Trapezoid Rule is a numerical method to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoid areas provides an estimate of the integral, improving accuracy as the number of subintervals increases.

Partitioning the Interval and Subintervals

To apply the Trapezoid Rule, the integration interval [a, b] is divided into n equal subintervals, each of width Δx = (b - a)/n. The function values at the endpoints and subinterval points are used to calculate the trapezoid areas, making the choice of n crucial for the approximation's precision.

Error Estimation in the Trapezoid Rule

The error bound for the Trapezoid Rule depends on the second derivative of the function over the interval. Theorem 8.1 provides a formula to estimate the maximum error, helping to understand how close the approximation is to the true integral and guiding the choice of n for desired accuracy.