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 0 to 2 of (t³ + t) dt

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First, write down the integral to be evaluated: \(\int_0^2 (t^3 + t) \, dt\).

To evaluate the integral directly, find the antiderivative of the integrand. Recall that the antiderivative of \(t^3\) is \(\frac{t^4}{4}\) and the antiderivative of \(t\) is \(\frac{t^2}{2}\).

Express the antiderivative as \(F(t) = \frac{t^4}{4} + \frac{t^2}{2}\).

Apply the Fundamental Theorem of Calculus by evaluating \(F(t)\) at the upper limit and subtracting the value at the lower limit: \(F(2) - F(0)\).

To find the error bound \(|E_T|\) for the Trapezoidal Rule, use the formula \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq t \leq b} |f''(t)|\), where \(f(t) = t^3 + t\). Compute the second derivative \(f''(t)\), find its maximum on \([0, 2]\), and substitute all values into the formula.

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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 the integral of t³ + t from 0 to 2, one finds the antiderivative, evaluates it at the bounds, and subtracts to get the exact area under the curve.

Error Bound for the Trapezoidal Rule (|ET|)

The error bound |ET| for the Trapezoidal Rule estimates the maximum difference between the approximate and exact integral values. It depends on the second derivative of the function and the width of the subintervals, providing a way to assess 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 0 to 2 of (t³ + t) dt