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 0 to 2 of sin(x + 1) dx

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Identify the function to be integrated: \(f(x) = \sin(x + 1)\) over the interval \([0, 2]\).

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

Compute the second derivative of the function: \[f'(x) = \cos(x + 1),\] \[f''(x) = -\sin(x + 1).\]

Set the error bound less than \(10^{-4}\) and solve for \(n\): \[\frac{(2 - 0)^3}{12 n^2} \times 1 < 10^{-4} \implies \frac{8}{12 n^2} < 10^{-4} \implies n^2 > \frac{8}{12 \times 10^{-4}}.\] From here, solve for \(n\) to find the minimum number of subintervals needed.

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Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the integration 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 integrand. Specifically, the error magnitude is at most (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 determine how many subintervals are needed to achieve a desired accuracy.

Second Derivative and Its Role in Error Estimation

The second derivative of the function indicates the concavity and affects the curvature of the graph. In error estimation for the Trapezoidal Rule, the maximum absolute value of the second derivative over the interval controls the error size. Calculating or bounding this value is essential to apply the error formula and find the minimum n.

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 0 to 2 of sin(x + 1) dx