Use Simpson's Rule with n = 10 to estimate the arc length of the curve. (Round your answer to six decimal places.) y = In(1 + x³), 0≤x≤ 5 Find the answer produced by a calculator or computer to compare with the previous result. (Round your answer to six decimal places.)
Use Simpson's Rule with n = 10 to estimate the arc length of the curve. (Round your answer to six decimal places.) y = In(1 + x³), 0≤x≤ 5 Find the answer produced by a calculator or computer to compare with the previous result. (Round your answer to six decimal places.)
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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![### Estimating Arc Length Using Simpson's Rule
#### Introduction
Simpson's Rule is a numerical method that can provide an estimated solution for the definite integral of a function. It is particularly used when an analytical solution is difficult or impossible to determine. This problem involves estimating the arc length of a given curve using Simpson's Rule.
#### Problem Statement
You are given the function \( y = \ln(1 + x^3) \) over the interval \( 0 \leq x \leq 5 \). The task requires using Simpson's Rule with \( n = 10 \) to estimate the arc length of the curve.
#### Steps
1. **Function Definition:**
\[
y = \ln(1 + x^3)
\]
with the interval \( 0 \leq x \leq 5 \).
2. **Using Simpson’s Rule:**
Use \( n = 10 \) to estimate the arc length.
3. **Compare with Computational Tools:**
Find the answer produced by a graphing calculator or computer and round your answer to six decimal places for both the Simpson's Rule estimate and the computational result.
#### Detailed Solution
1. **Applying Simpson's Rule:**
Simpson's Rule requires dividing the interval into \( n \) subintervals of equal width.
2. **Arc Length Formula:**
The arc length \( L \) of the curve \( y = f(x) \) from \( x = a \) to \( x = b \) can be approximated using Simpson's Rule in the following manner:
\[
L \approx \frac{\Delta x}{3} \left( f(a) + 4f(a+\Delta x) + 2f(a+2\Delta x) + \ldots + 4f(b-\Delta x) + f(b) \right)
\]
Here, \( \Delta x = \frac{b - a}{n} \). In this case, \( a = 0 \), \( b = 5 \), and \( n = 10 \).
3. **Graphing Calculator or Computer Estimate:**
Compare this result with a more precise value obtained via a graphing calculator or computer software.
#### Answer Fields
- **Simpson's Rule Estimation:**
\[
\text{(Rounded to six decimal places)}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06f55c03-0ff2-4f1b-a27d-6e1bf7c58dcc%2Ffe340030-7f18-44c2-aca1-210c89e1a863%2Fhhpdncv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Estimating Arc Length Using Simpson's Rule
#### Introduction
Simpson's Rule is a numerical method that can provide an estimated solution for the definite integral of a function. It is particularly used when an analytical solution is difficult or impossible to determine. This problem involves estimating the arc length of a given curve using Simpson's Rule.
#### Problem Statement
You are given the function \( y = \ln(1 + x^3) \) over the interval \( 0 \leq x \leq 5 \). The task requires using Simpson's Rule with \( n = 10 \) to estimate the arc length of the curve.
#### Steps
1. **Function Definition:**
\[
y = \ln(1 + x^3)
\]
with the interval \( 0 \leq x \leq 5 \).
2. **Using Simpson’s Rule:**
Use \( n = 10 \) to estimate the arc length.
3. **Compare with Computational Tools:**
Find the answer produced by a graphing calculator or computer and round your answer to six decimal places for both the Simpson's Rule estimate and the computational result.
#### Detailed Solution
1. **Applying Simpson's Rule:**
Simpson's Rule requires dividing the interval into \( n \) subintervals of equal width.
2. **Arc Length Formula:**
The arc length \( L \) of the curve \( y = f(x) \) from \( x = a \) to \( x = b \) can be approximated using Simpson's Rule in the following manner:
\[
L \approx \frac{\Delta x}{3} \left( f(a) + 4f(a+\Delta x) + 2f(a+2\Delta x) + \ldots + 4f(b-\Delta x) + f(b) \right)
\]
Here, \( \Delta x = \frac{b - a}{n} \). In this case, \( a = 0 \), \( b = 5 \), and \( n = 10 \).
3. **Graphing Calculator or Computer Estimate:**
Compare this result with a more precise value obtained via a graphing calculator or computer software.
#### Answer Fields
- **Simpson's Rule Estimation:**
\[
\text{(Rounded to six decimal places)}
\]
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