The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
II. Using the Trapezoidal Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.
∫ from 0 to π of sin(t) dth

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Identify the integral to approximate: \(\int_0^{\pi} \sin(t) \, dt\) using the Trapezoidal Rule with \(n=4\) subintervals.

Calculate the step size \(h\) using the formula \(h = \frac{b - a}{n}\), where \(a=0\) and \(b=\pi\). So, \(h = \frac{\pi - 0}{4} = \frac{\pi}{4}\).

Determine the partition points: \(t_0 = 0\), \(t_1 = \frac{\pi}{4}\), \(t_2 = \frac{\pi}{2}\), \(t_3 = \frac{3\pi}{4}\), and \(t_4 = \pi\).

Apply the Trapezoidal Rule formula: \(T_n = \frac{h}{2} \left[f(t_0) + 2f(t_1) + 2f(t_2) + 2f(t_3) + f(t_4)\right]\) where \(f(t) = \sin(t)\).

To find an upper bound for the error \(|E_T|\), use the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq t \leq b} |f''(t)|\). Calculate \(f''(t)\) for \(f(t) = \sin(t)\), find its maximum absolute value on \([0, \pi]\), and substitute all values to get the error bound.

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주요 개념

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Trapezoidal Rule

The Trapezoidal 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 trapezoidal areas provides an estimate of the integral, improving accuracy with more subintervals (n).

Error Bound for the Trapezoidal Rule

The error bound for the Trapezoidal Rule estimates the maximum possible difference between the true integral and its approximation. It depends on the second derivative of the function, the interval length, and the number of subintervals, providing a way to assess the accuracy of the approximation.

Properties of the Sine Function on [0, π]

Understanding the behavior of sin(t) on [0, π] is crucial, as it is positive and smooth with known derivatives. Its second derivative, -sin(t), helps determine the error bound for the Trapezoidal Rule, since the maximum absolute value of this derivative affects the error estimate.

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.II. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.∫ from 0 to π of sin(t) dth