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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Question
![The image displays a definite integral, which is a fundamental concept in calculus. The integral is:
\[
\int_{1}^{3} x \sqrt{x^2 + 2x + 5} \, dx
\]
### Explanation:
- **Integral**: The symbol \(\int\) represents integration, which is the process of finding the area under a curve represented by a function.
- **Limits of Integration**: The numbers 1 and 3 at the bottom and top of the integral symbol indicate that it is a definite integral, calculated between the lower limit 1 and the upper limit 3.
- **Function**: The function to be integrated is \(x \sqrt{x^2 + 2x + 5}\). This includes:
- \(x\) is the variable of integration.
- \(\sqrt{x^2 + 2x + 5}\) is the square root of the quadratic expression inside the integral.
- **\(dx\)**: This indicates that the integration is with respect to \(x\).
This integral calculates the area under the curve of the given function from x = 1 to x = 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccac55ee-732b-4607-a38e-f5bd0a8aacb2%2Ff5007e34-b684-433c-8275-368e6b7fb850%2Fmj8k3jb_processed.png&w=3840&q=75)
Transcribed Image Text:The image displays a definite integral, which is a fundamental concept in calculus. The integral is:
\[
\int_{1}^{3} x \sqrt{x^2 + 2x + 5} \, dx
\]
### Explanation:
- **Integral**: The symbol \(\int\) represents integration, which is the process of finding the area under a curve represented by a function.
- **Limits of Integration**: The numbers 1 and 3 at the bottom and top of the integral symbol indicate that it is a definite integral, calculated between the lower limit 1 and the upper limit 3.
- **Function**: The function to be integrated is \(x \sqrt{x^2 + 2x + 5}\). This includes:
- \(x\) is the variable of integration.
- \(\sqrt{x^2 + 2x + 5}\) is the square root of the quadratic expression inside the integral.
- **\(dx\)**: This indicates that the integration is with respect to \(x\).
This integral calculates the area under the curve of the given function from x = 1 to x = 3.
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