1 f (x) %3D 3 x2 Find the Riemann sum of f (x) over the interval 1, 31, using its ten subintervals of equal length, and using the left- end numbers as the representative numbers.
1 f (x) %3D 3 x2 Find the Riemann sum of f (x) over the interval 1, 31, using its ten subintervals of equal length, and using the left- end numbers as the representative numbers.
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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![### Riemann Sum Calculation for Function f(x)
Given the function:
\[ f(x) = \frac{1}{3x^2} \]
We aim to find the Riemann sum of \( f(x) \) over the interval \([1, 31]\), using **ten** subintervals of equal length and the **left-end numbers** as the representative points.
#### Steps:
1. **Determine the Subinterval Length:**
The interval \([1, 31]\) is divided into 10 subintervals. The length of each subinterval, \(\Delta x\), can be calculated as:
\[
\Delta x = \frac{31 - 1}{10} = 3
\]
2. **Identify the Left-End Points:**
The left-end points for each subinterval are:
\[
x_0 = 1, \quad x_1 = 4, \quad x_2 = 7, \quad x_3 = 10, \quad x_4 = 13, \quad x_5 = 16, \quad x_6 = 19, \quad x_7 = 22, \quad x_8 = 25, \quad x_9 = 28
\]
3. **Apply the Function to Each Left-End Point:**
Evaluate the function at each left-end point:
\[
f(x_0) = f(1), \quad f(x_1) = f(4), \quad f(x_2) = f(7), \quad \ldots, \quad f(x_9) = f(28)
\]
4. **Construct the Riemann Sum:**
The Riemann sum \( R \) is given by:
\[
R = \sum_{i=0}^{9} f(x_i) \Delta x
\]
5. **Illustrate the Subintervals on a Number Line:**
A number line is provided to represent the subintervals.
\[
\begin{array}{ccccccccccc}
1 & \mid & 4 & \mid & 7 & \mid & 10 & \mid & 13 & \mid & 16 & \mid & 19 & \mid & 22 & \mid & 25 &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F239df334-5b03-4bc0-9001-d398335a6cd3%2Ffc158026-fc71-4106-948a-466cc35fa815%2F5j23i7_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Riemann Sum Calculation for Function f(x)
Given the function:
\[ f(x) = \frac{1}{3x^2} \]
We aim to find the Riemann sum of \( f(x) \) over the interval \([1, 31]\), using **ten** subintervals of equal length and the **left-end numbers** as the representative points.
#### Steps:
1. **Determine the Subinterval Length:**
The interval \([1, 31]\) is divided into 10 subintervals. The length of each subinterval, \(\Delta x\), can be calculated as:
\[
\Delta x = \frac{31 - 1}{10} = 3
\]
2. **Identify the Left-End Points:**
The left-end points for each subinterval are:
\[
x_0 = 1, \quad x_1 = 4, \quad x_2 = 7, \quad x_3 = 10, \quad x_4 = 13, \quad x_5 = 16, \quad x_6 = 19, \quad x_7 = 22, \quad x_8 = 25, \quad x_9 = 28
\]
3. **Apply the Function to Each Left-End Point:**
Evaluate the function at each left-end point:
\[
f(x_0) = f(1), \quad f(x_1) = f(4), \quad f(x_2) = f(7), \quad \ldots, \quad f(x_9) = f(28)
\]
4. **Construct the Riemann Sum:**
The Riemann sum \( R \) is given by:
\[
R = \sum_{i=0}^{9} f(x_i) \Delta x
\]
5. **Illustrate the Subintervals on a Number Line:**
A number line is provided to represent the subintervals.
\[
\begin{array}{ccccccccccc}
1 & \mid & 4 & \mid & 7 & \mid & 10 & \mid & 13 & \mid & 16 & \mid & 19 & \mid & 22 & \mid & 25 &
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