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 x dx

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Identify the integral to evaluate: \(\int_{1}^{2} x \, dx\).

Evaluate the integral directly by finding the antiderivative of the function \(f(x) = x\). The antiderivative is \(F(x) = \frac{1}{2}x^{2}\).

Apply the Fundamental Theorem of Calculus by computing \(F(2) - F(1)\), which gives the exact value of the integral.

Recall the formula for the Trapezoidal Rule error bound \(|E_{T}| \leq \frac{(b - a)^{3}}{12n^{2}} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals, \(a=1\), and \(b=2\).

Calculate the second derivative \(f''(x)\) of the function \(f(x) = x\), which is zero, and use this to find the exact error bound \(|E_{T}|\) for the Trapezoidal Rule.

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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 area under a curve into trapezoids rather than rectangles. It estimates the integral by averaging the function values at the endpoints of subintervals and multiplying by the subinterval width, providing a better approximation than simple Riemann sums.

Exact Evaluation of Definite Integrals

Exact evaluation involves finding the precise value of a definite integral using antiderivatives and the Fundamental Theorem of Calculus. For ∫ from 1 to 2 of x dx, the antiderivative of x is (1/2)x², and evaluating it at the bounds gives the exact area under the curve.

Error Estimation in Numerical Integration (|ET|)

Error estimation quantifies the difference between the exact integral value and its numerical approximation. For the Trapezoidal Rule, the error bound |ET| depends on the second derivative of the function and the width of the subintervals, helping assess 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 x dx