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
b. Evaluate the integral directly and find |ET|.
∫ from 1 to 2 of 1 / s² ds

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First, identify the integral to evaluate: \(\int_{1}^{2} \frac{1}{s^{2}} \, ds\).

To evaluate the integral directly, rewrite the integrand as \(s^{-2}\) and find its antiderivative. Recall that the antiderivative of \(s^{n}\) is \(\frac{s^{n+1}}{n+1}\) for \(n \neq -1\).

Compute the antiderivative: \(\int s^{-2} \, ds = \frac{s^{-1}}{-1} = -s^{-1} = -\frac{1}{s}\).

Evaluate the definite integral by applying the Fundamental Theorem of Calculus: calculate \(-\frac{1}{s}\) at the upper limit \(s=2\) and subtract its value at the lower limit \(s=1\).

To find the error bound \(|E_{T}|\) for the Trapezoidal Rule, use the formula \(|E_{T}| \leq \frac{(b - a)^{3}}{12 n^{2}} \max_{a \leq s \leq b} |f''(s)|\), where \(f(s) = \frac{1}{s^{2}}\). Compute the second derivative \(f''(s)\), find its maximum absolute value on \([1,2]\), and substitute all values into the formula.

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주요 개념

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Definite Integral

A definite integral calculates the exact area under a curve between two limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the interval bounds. Evaluating the integral directly involves finding the antiderivative and applying the Fundamental Theorem of Calculus.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the area under the curve into trapezoids. It estimates the integral by summing the areas of these trapezoids, which is useful when the integral is difficult to evaluate analytically.

Trapezoidal Rule Error Bound (|ET|)

The error bound |ET| for the Trapezoidal Rule estimates the maximum difference between the exact integral and its trapezoidal approximation. It depends on the second derivative of the function and the width of the subintervals, providing insight into the accuracy of the approximation.

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 Ruleb. Evaluate the integral directly and find |ET|.∫ from 1 to 2 of 1 / s² ds