2. Use the Trope zoid to estimate the' valve the integral numerically. Give your final se the Trope zoid Rule with n=8 subinterals pe of rounded to three deama ans wer l places. For matimom precision, do not round Show until the end. our wock, 3. x、十X xp
2. Use the Trope zoid to estimate the' valve the integral numerically. Give your final se the Trope zoid Rule with n=8 subinterals pe of rounded to three deama ans wer l places. For matimom precision, do not round Show until the end. our wock, 3. x、十X xp
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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![### Numerical Integration using the Trapezoidal Rule
#### Problem Statement:
Use the Trapezoid Rule with \( n = 8 \) subintervals to estimate the value of the integral numerically. Give your final answer rounded to three decimal places. For maximum precision, do not round until the end. Show your work.
\[
\int_{0}^{8} \sqrt{x^3 + x} \, dx
\]
#### Explanation:
The Trapezoidal Rule is a numerical integration method used to approximate the definite integral of a function. When using this method, we divide the integration interval into smaller subintervals (trapezoids), calculate the area of each trapezoid, and sum these areas to approximate the integral.
### Steps:
1. Identify the given integral and its limits:
\[
\int_{0}^{8} \sqrt{x^3 + x} \, dx
\]
2. Determine the number of subintervals (n) and calculate the width (h) of each subinterval:
\[
n = 8
\]
\[
h = \frac{b - a}{n} = \frac{8 - 0}{8} = 1
\]
3. Apply the Trapezoidal Rule formula:
\[
\int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]
\]
where \( a = 0 \), \( b = 8 \), and \( x_i = a + i \cdot h \).
### Calculation:
4. Compute \( f(x_i) \) for all subintervals:
\[
x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3, \quad x_4 = 4, \quad x_5 = 5, \quad x_6 = 6, \quad x_7 = 7, \quad x_8 = 8
\]
\[
f(x) = \sqrt{x^3 + x}
\]
Therefore,
\[
f(x_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bf7da54-5138-4b57-a0f9-aec30fa8e8cf%2F9758e033-6627-46ae-89bb-ad14c2ad107a%2F8brsm1e.jpeg&w=3840&q=75)
Transcribed Image Text:### Numerical Integration using the Trapezoidal Rule
#### Problem Statement:
Use the Trapezoid Rule with \( n = 8 \) subintervals to estimate the value of the integral numerically. Give your final answer rounded to three decimal places. For maximum precision, do not round until the end. Show your work.
\[
\int_{0}^{8} \sqrt{x^3 + x} \, dx
\]
#### Explanation:
The Trapezoidal Rule is a numerical integration method used to approximate the definite integral of a function. When using this method, we divide the integration interval into smaller subintervals (trapezoids), calculate the area of each trapezoid, and sum these areas to approximate the integral.
### Steps:
1. Identify the given integral and its limits:
\[
\int_{0}^{8} \sqrt{x^3 + x} \, dx
\]
2. Determine the number of subintervals (n) and calculate the width (h) of each subinterval:
\[
n = 8
\]
\[
h = \frac{b - a}{n} = \frac{8 - 0}{8} = 1
\]
3. Apply the Trapezoidal Rule formula:
\[
\int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]
\]
where \( a = 0 \), \( b = 8 \), and \( x_i = a + i \cdot h \).
### Calculation:
4. Compute \( f(x_i) \) for all subintervals:
\[
x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3, \quad x_4 = 4, \quad x_5 = 5, \quad x_6 = 6, \quad x_7 = 7, \quad x_8 = 8
\]
\[
f(x) = \sqrt{x^3 + x}
\]
Therefore,
\[
f(x_
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