### Numerical Integration Techniques - Error Estimation#### Problem StatementEstimate the minimum number of subintervals to approximate the value of\[ \int_{-2}^{2} (4x^2 + 4) \, dx \]with an error of magnitude less than $3 \times 10^{-4}$ using:a. the error estimate formula for the Trapezoidal Rule.b. the error estimate formula for Simpson’s Rule.#### Explanation:### Part a: Error Estimate Formula for the Trapezoidal RuleThe error estimate for the Trapezoidal Rule is given by:\[ E_T \leq \frac{(b-a)^3}{12n^2} \max_{a \leq x \leq b} |f''(x)| \]1. Identify $ f(x) $: For $ f(x) = 4x^2 + 4 $2. Compute the second derivative $ f''(x) $: \[ f''(x) = 8 \]3. Calculate the bounds: \[ a = -2 \] \[ b = 2 \]4. Substitute into the error formula: \[ E_T \leq \frac{(2 - (-2))^3}{12n^2} (8) \] \[ E_T \leq \frac{64}{12n^2} \times 8 \] \[ E_T \leq \frac{512}{12n^2} \] \[ E_T \leq \frac{128}{3n^2} \]5. Set up the inequality: \[ \frac{128}{3n^2} \leq 3 \times 10^{-4} \]6. Solve for $ n $: \[ n \geq \sqrt{\frac{128}{3 \times 3 \times 10^{-4}}} \] \[ n \geq \sqrt{\frac{128}{9 \times 10^{-4}}} \] \[ n \geq \sqrt{142222.22} \] \[ n \geq 377 \]Therefore, the minimum number of subintervals $ n $ must be greater than or equal to 377 for the Trapezoidal Rule.

We will solve both these parts using basics of numerical analysis.

2 Estimate the minimum number of subintervals to approximate the value of| (4x² + 4) dx with an error of magnitude less than 3x 10-* using -2 a. the error estimate formula for the Trapezoidal Rule. b. the error estimate formula for Simpson's Rule. N SY

2 Estimate the minimum number of subintervals to approximate the value of| (4x² + 4) dx with an error of magnitude less than 3x 10-* using -2 a. the error estimate formula for the Trapezoidal Rule. b. the error estimate formula for Simpson's Rule. N SY

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differential equations. The most important aspect of this theory is error analysis.

Question

### Numerical Integration Techniques - Error Estimation

#### Problem Statement

Estimate the minimum number of subintervals to approximate the value of
\[ \int_{-2}^{2} (4x^2 + 4) \, dx \]
with an error of magnitude less than $3 \times 10^{-4}$ using:
a. the error estimate formula for the Trapezoidal Rule.
b. the error estimate formula for Simpson’s Rule.

#### Explanation:

### Part a: Error Estimate Formula for the Trapezoidal Rule

The error estimate for the Trapezoidal Rule is given by:
\[ E_T \leq \frac{(b-a)^3}{12n^2} \max_{a \leq x \leq b} |f''(x)| \]

1. Identify $ f(x) $:
For $ f(x) = 4x^2 + 4 $

2. Compute the second derivative $ f''(x) $:
\[ f''(x) = 8 \]

3. Calculate the bounds:
\[ a = -2 \]
\[ b = 2 \]

4. Substitute into the error formula:
\[ E_T \leq \frac{(2 - (-2))^3}{12n^2} (8) \]
\[ E_T \leq \frac{64}{12n^2} \times 8 \]
\[ E_T \leq \frac{512}{12n^2} \]
\[ E_T \leq \frac{128}{3n^2} \]

5. Set up the inequality:
\[ \frac{128}{3n^2} \leq 3 \times 10^{-4} \]

6. Solve for $ n $:
\[ n \geq \sqrt{\frac{128}{3 \times 3 \times 10^{-4}}} \]
\[ n \geq \sqrt{\frac{128}{9 \times 10^{-4}}} \]
\[ n \geq \sqrt{142222.22} \]
\[ n \geq 377 \]

Therefore, the minimum number of subintervals $ n $ must be greater than or equal to 377 for the Trapezoidal Rule.

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