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 -2 to 0 of (x² - 1) dx

검증된 단계별 안내

First, write down the integral to be evaluated directly: \(\int_{-2}^{0} (x^{2} - 1) \, dx\).

Next, find the antiderivative of the integrand \(x^{2} - 1\). Recall that the antiderivative of \(x^{2}\) is \(\frac{x^{3}}{3}\) and the antiderivative of \(-1\) is \(-x\). So, the antiderivative is \(F(x) = \frac{x^{3}}{3} - x\).

Evaluate the antiderivative at the upper and lower limits of integration: calculate \(F(0)\) and \(F(-2)\).

Subtract the values to find the exact value of the integral: \(\int_{-2}^{0} (x^{2} - 1) \, dx = F(0) - F(-2)\).

To find the error bound \(|E_{T}|\) for the Trapezoidal Rule, recall the error formula: \(|E_{T}| \leq \frac{(b - a)^{3}}{12 n^{2}} \max_{a \leq x \leq b} |f''(x)|\), where \(a = -2\), \(b = 0\), and \(n\) is the number of subintervals used. Compute \(f''(x)\), find its maximum absolute value on \([-2,0]\), 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 it 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 exact integral is difficult to compute.

Error Bound for the Trapezoidal Rule (|ET|)

The error bound |ET| measures the difference between the exact integral and the Trapezoidal Rule approximation. It depends on the second derivative of the function and the number of subintervals, providing a way to estimate the accuracy of the numerical 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 -2 to 0 of (x² - 1) dx