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 Rectangle Method
#### Problem Statement:
Approximate the integral
\[ \int_{0}^{3} \frac{x}{x^2 + 2} \, dx \]
using 6 rectangles.
#### Instruction:
Calculate the approximate value and round your final answer to 3 decimal places. Do NOT round until the end of the calculation.
#### Detailed Steps:
1. **Function to Integrate:**
\[ f(x) = \frac{x}{x^2 + 2} \]
2. **Interval:**
From 0 to 3.
3. **Number of Rectangles:**
6 rectangles.
4. **Width of Each Rectangle (Δx):**
The width Δx is calculated as:
\[ \Delta x = \frac{b - a}{n} = \frac{3 - 0}{6} = 0.5 \]
5. **Rectangle Heights:**
Heights are determined by evaluating the function \( f(x) \) at the endpoints or midpoints of each subinterval, depending on the chosen method (left, right, or midpoint).
#### Example for Midpoint Rule:
If we choose the midpoint (this will be more accurate):
\[ x_i^* = 0.25, 0.75, 1.25, 1.75, 2.25, 2.75 \]
#### Evaluate \( f(x) \) at Midpoints:
\[ f(0.25) = \frac{0.25}{(0.25)^2 + 2} = \frac{0.25}{0.0625 + 2} = \frac{0.25}{2.0625} \]
\[ f(0.75) = \frac{0.75}{0.75^2 + 2} = \frac{0.75}{0.5625 + 2} = \frac{0.75}{2.5625} \]
\[ f(1.25) = \frac{1.25}{(1.25)^2 + 2} = \frac{1.25}{1.5625 + 2} = \frac{1.25}{3.5625} \]
\[ f(1.75) = \frac{1.75}{(1.75)^2 + 2} = \frac{1.75}{3.0625](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f26cfff-a5a0-4f1a-b4dd-ba875949e336%2Fa3924243-561c-4745-af22-190b17cf6a71%2Fx77b8dl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Numerical Integration Using the Rectangle Method
#### Problem Statement:
Approximate the integral
\[ \int_{0}^{3} \frac{x}{x^2 + 2} \, dx \]
using 6 rectangles.
#### Instruction:
Calculate the approximate value and round your final answer to 3 decimal places. Do NOT round until the end of the calculation.
#### Detailed Steps:
1. **Function to Integrate:**
\[ f(x) = \frac{x}{x^2 + 2} \]
2. **Interval:**
From 0 to 3.
3. **Number of Rectangles:**
6 rectangles.
4. **Width of Each Rectangle (Δx):**
The width Δx is calculated as:
\[ \Delta x = \frac{b - a}{n} = \frac{3 - 0}{6} = 0.5 \]
5. **Rectangle Heights:**
Heights are determined by evaluating the function \( f(x) \) at the endpoints or midpoints of each subinterval, depending on the chosen method (left, right, or midpoint).
#### Example for Midpoint Rule:
If we choose the midpoint (this will be more accurate):
\[ x_i^* = 0.25, 0.75, 1.25, 1.75, 2.25, 2.75 \]
#### Evaluate \( f(x) \) at Midpoints:
\[ f(0.25) = \frac{0.25}{(0.25)^2 + 2} = \frac{0.25}{0.0625 + 2} = \frac{0.25}{2.0625} \]
\[ f(0.75) = \frac{0.75}{0.75^2 + 2} = \frac{0.75}{0.5625 + 2} = \frac{0.75}{2.5625} \]
\[ f(1.25) = \frac{1.25}{(1.25)^2 + 2} = \frac{1.25}{1.5625 + 2} = \frac{1.25}{3.5625} \]
\[ f(1.75) = \frac{1.75}{(1.75)^2 + 2} = \frac{1.75}{3.0625
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