### Numerical Integration: Trapezoidal Rule and Simpson's RuleEvaluate the integral:\[ \int_{2}^{8} \frac{4}{s^2} \, ds \]using the trapezoidal rule and Simpson's rule.#### Steps to Determine:1. **The value of the integral directly**: Evaluate the integral analytically to find the exact value.2. **The trapezoidal rule estimate for $ n = 4 $**: Divide the interval $[2, 8]$ into 4 equal sub-intervals and apply the trapezoidal rule formula: \[ T = \frac{b - a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(a + ih) + f(b) \right] \] where $ h = \frac{b - a}{n} $.3. **An upper bound for $ | E_T | $**: Use the error bound formula for the trapezoidal rule: \[ | E_T | \leq \frac{(b - a)^3}{12n^2} M \] Here, $ M $ is the maximum value of $ |f''(x)| $ on $[a, b]$.4. **The upper bound for $ | E_T | $ as a percentage of the integral's true value**: Calculate $ \left| E_T \right| $ as a fraction of the true value of the integral, and then convert it to a percentage.5. **The Simpson's rule estimate for $ n = 4 $**: Apply Simpson’s rule: \[ S = \frac{b - a}{3n} \left[ f(a) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(b) \right] \]6. **An upper bound for $ | E_S | $**: Use the error bound formula for Simpson’s rule: \[ | E_S | \leq \frac{(b - a)^5}{180n^4} M \] Here, $ M $ is the maximum value

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8 Evaluate ds using the trapezoidal rule and Simpson's rule. Determine i. the value of the integral directly. ii. the trapezoidal rule estimate forn=4. iii. an upper bound for |E-|- iv. the upper bound for |E-| as a percentage of the integral's true value. v. the Simpson's rule estimate for n= 4. vi. an upper bound for Es vii. the upper bound for Es as a percentage of the integral's true value.

8 Evaluate ds using the trapezoidal rule and Simpson's rule. Determine i. the value of the integral directly. ii. the trapezoidal rule estimate forn=4. iii. an upper bound for |E-|- iv. the upper bound for |E-| as a percentage of the integral's true value. v. the Simpson's rule estimate for n= 4. vi. an upper bound for Es vii. the upper bound for Es as a percentage of the integral's true value.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Numerical Integration: Trapezoidal Rule and Simpson's Rule

Evaluate the integral:

\[ \int_{2}^{8} \frac{4}{s^2} \, ds \]

using the trapezoidal rule and Simpson's rule.

#### Steps to Determine:
1. **The value of the integral directly**:
Evaluate the integral analytically to find the exact value.

2. **The trapezoidal rule estimate for $ n = 4 $**:
Divide the interval $[2, 8]$ into 4 equal sub-intervals and apply the trapezoidal rule formula:

\[
T = \frac{b - a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(a + ih) + f(b) \right]
\]

where $ h = \frac{b - a}{n} $.

3. **An upper bound for $ | E_T | $**:
Use the error bound formula for the trapezoidal rule:

\[
| E_T | \leq \frac{(b - a)^3}{12n^2} M
\]

Here, $ M $ is the maximum value of $ |f''(x)| $ on $[a, b]$.

4. **The upper bound for $ | E_T | $ as a percentage of the integral's true value**:
Calculate $ \left| E_T \right| $ as a fraction of the true value of the integral, and then convert it to a percentage.

5. **The Simpson's rule estimate for $ n = 4 $**:
Apply Simpson’s rule:

\[
S = \frac{b - a}{3n} \left[ f(a) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(b) \right]
\]

6. **An upper bound for $ | E_S | $**:
Use the error bound formula for Simpson’s rule:

\[
| E_S | \leq \frac{(b - a)^5}{180n^4} M
\]

Here, $ M $ is the maximum value

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