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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![**Problem Statement**
Find the sum of the values of \( f(x) = x^3 \) over the integers \( 1, 2, 3, \ldots, 10 \).
**Solution**
To solve this problem, we need to calculate the sum of the cubes of the integers from 1 to 10. Specifically, we need to find:
\[
f(1) + f(2) + f(3) + \cdots + f(10)
\]
Given the function \( f(x) = x^3 \), substitute \( x \) with integers from 1 to 10 and sum them up:
\[
1^3 + 2^3 + 3^3 + \cdots + 10^3
\]
First, we calculate each cube:
\[
\begin{align*}
1^3 & = 1 \\
2^3 & = 8 \\
3^3 & = 27 \\
4^3 & = 64 \\
5^3 & = 125 \\
6^3 & = 216 \\
7^3 & = 343 \\
8^3 & = 512 \\
9^3 & = 729 \\
10^3 & = 1000 \\
\end{align*}
\]
Next, sum these values:
\[
1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000
\]
\[
= 3025
\]
So, the sum of the values of \( f(x) = x^3 \) over the integers from 1 to 10 is \( 3025 \).
Therefore, the answer is:
\[
\boxed{3025}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11c3e735-47a2-4be2-bcf0-76944b0744ea%2Ff82ee54e-a282-4744-b189-f7fe2dac7128%2Fcyxmrqc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the sum of the values of \( f(x) = x^3 \) over the integers \( 1, 2, 3, \ldots, 10 \).
**Solution**
To solve this problem, we need to calculate the sum of the cubes of the integers from 1 to 10. Specifically, we need to find:
\[
f(1) + f(2) + f(3) + \cdots + f(10)
\]
Given the function \( f(x) = x^3 \), substitute \( x \) with integers from 1 to 10 and sum them up:
\[
1^3 + 2^3 + 3^3 + \cdots + 10^3
\]
First, we calculate each cube:
\[
\begin{align*}
1^3 & = 1 \\
2^3 & = 8 \\
3^3 & = 27 \\
4^3 & = 64 \\
5^3 & = 125 \\
6^3 & = 216 \\
7^3 & = 343 \\
8^3 & = 512 \\
9^3 & = 729 \\
10^3 & = 1000 \\
\end{align*}
\]
Next, sum these values:
\[
1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000
\]
\[
= 3025
\]
So, the sum of the values of \( f(x) = x^3 \) over the integers from 1 to 10 is \( 3025 \).
Therefore, the answer is:
\[
\boxed{3025}
\]
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