In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 1 to 3 of (2x - 1) dx

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Identify the function to be integrated: \(f(x) = 2x - 1\) over the interval \([1, 3]\).

Recall the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals.

Calculate the second derivative of the function: \(f''(x) = \frac{d^2}{dx^2}(2x - 1)\). Since \(f(x)\) is linear, \(f''(x) = 0\) for all \(x\).

Since \(f''(x) = 0\), the error bound formula implies the Trapezoidal Rule will give the exact value regardless of \(n\), so the minimum number of subintervals needed to achieve an error less than \(10^{-4}\) is 1.

Conclude that for this linear function, the Trapezoidal Rule is exact with just one subinterval, so no further subdivision is necessary.

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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 estimates the integral, with accuracy improving as the number of subintervals increases.

Error Bound for the Trapezoidal Rule

The error bound for the Trapezoidal Rule depends on the second derivative of the function being integrated. It is given by |E| ≤ (K(b - a)^3) / (12n^2), where K is the maximum absolute value of the second derivative on [a, b], and n is the number of subintervals. This formula helps estimate how many subintervals are needed to achieve a desired accuracy.

Second Derivative and Its Role in Error Estimation

The second derivative of the integrand measures the function's concavity and affects the error in the Trapezoidal Rule. A smaller maximum second derivative on the interval means less curvature and thus a smaller error, allowing fewer subintervals for a given accuracy.

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)∫ from 1 to 3 of (2x - 1) dx