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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![The image depicts a definite integral, written in mathematical notation as:
\[
\int_{1}^{3} (3x^2 - 2x + 1) \, dx
\]
This represents the integral of the function \(3x^2 - 2x + 1\) with respect to \(x\), evaluated over the interval from \(x = 1\) to \(x = 3\).
Detailed Explanation:
1. **Integral Sign \(\int\):** Indicates integration, a fundamental concept in calculus used to find the area under a curve or the accumulation of quantities.
2. **Limits of Integration:** The numbers 1 and 3 at the bottom and top of the integral sign denote the lower and upper limits of integration, respectively. This means we calculate the integral of the function from \(x = 1\) to \(x = 3\).
3. **Function to Integrate \((3x^2 - 2x + 1)\):** The polynomial function presented within the parentheses is the function being integrated. It is a quadratic polynomial composed of three terms:
- \(3x^2\) is a term that represents a parabola when graphed.
- \(-2x\) is a linear term that tilts the parabola.
- \(+1\) is a constant term that vertically shifts the graph.
4. **\(dx\):** This notation signifies that the integration is with respect to the variable \(x\).
This integral can be solved using standard techniques in calculus to find the area under the curve described by the polynomial between \(x = 1\) and \(x = 3\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e8eaad3-4d91-4777-be35-ba41f897f0fa%2F6519d63b-34b0-4224-9495-7e15f729d014%2Fhsv6un_processed.png&w=3840&q=75)
Transcribed Image Text:The image depicts a definite integral, written in mathematical notation as:
\[
\int_{1}^{3} (3x^2 - 2x + 1) \, dx
\]
This represents the integral of the function \(3x^2 - 2x + 1\) with respect to \(x\), evaluated over the interval from \(x = 1\) to \(x = 3\).
Detailed Explanation:
1. **Integral Sign \(\int\):** Indicates integration, a fundamental concept in calculus used to find the area under a curve or the accumulation of quantities.
2. **Limits of Integration:** The numbers 1 and 3 at the bottom and top of the integral sign denote the lower and upper limits of integration, respectively. This means we calculate the integral of the function from \(x = 1\) to \(x = 3\).
3. **Function to Integrate \((3x^2 - 2x + 1)\):** The polynomial function presented within the parentheses is the function being integrated. It is a quadratic polynomial composed of three terms:
- \(3x^2\) is a term that represents a parabola when graphed.
- \(-2x\) is a linear term that tilts the parabola.
- \(+1\) is a constant term that vertically shifts the graph.
4. **\(dx\):** This notation signifies that the integration is with respect to the variable \(x\).
This integral can be solved using standard techniques in calculus to find the area under the curve described by the polynomial between \(x = 1\) and \(x = 3\).
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