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: Evaluating the Integral**
Evaluate the integral by interpreting it in terms of areas.
\[
\int_{-1}^{2} (1 - x) \, dx
\]
*A blank space is provided for the solution.*
**Explanation for Educational Context:**
This problem involves finding the definite integral of the function \( f(x) = 1 - x \) over the interval \([-1, 2]\). The integral represents the net area between the curve \( y = 1 - x \) and the x-axis from \( x = -1 \) to \( x = 2 \).
**Steps for Solving:**
1. **Identify the Function:** The function \( f(x) = 1 - x \) is a linear function with a y-intercept of 1 and a slope of -1.
2. **Visualize the Graph:**
- The line intersects the y-axis at \( (0, 1) \).
- It intersects the x-axis at \( (1, 0) \).
3. **Determine the Area:**
- **Region 1:** From \( x = -1 \) to \( x = 1 \), the area is a trapezoid (or the combination of a rectangle and a triangle).
- **Region 2:** From \( x = 1 \) to \( x = 2 \), the area is a triangle below the x-axis, which contributes negatively to the total area.
To solve the integral, you'll need to find these areas and sum them, keeping in mind that areas below the x-axis are subtracted from the total.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a15759c-8b82-438f-9d6a-36f728df4f17%2Fa79be227-5810-4e85-892d-d68cb0340376%2Fuckj2lw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Evaluating the Integral**
Evaluate the integral by interpreting it in terms of areas.
\[
\int_{-1}^{2} (1 - x) \, dx
\]
*A blank space is provided for the solution.*
**Explanation for Educational Context:**
This problem involves finding the definite integral of the function \( f(x) = 1 - x \) over the interval \([-1, 2]\). The integral represents the net area between the curve \( y = 1 - x \) and the x-axis from \( x = -1 \) to \( x = 2 \).
**Steps for Solving:**
1. **Identify the Function:** The function \( f(x) = 1 - x \) is a linear function with a y-intercept of 1 and a slope of -1.
2. **Visualize the Graph:**
- The line intersects the y-axis at \( (0, 1) \).
- It intersects the x-axis at \( (1, 0) \).
3. **Determine the Area:**
- **Region 1:** From \( x = -1 \) to \( x = 1 \), the area is a trapezoid (or the combination of a rectangle and a triangle).
- **Region 2:** From \( x = 1 \) to \( x = 2 \), the area is a triangle below the x-axis, which contributes negatively to the total area.
To solve the integral, you'll need to find these areas and sum them, keeping in mind that areas below the x-axis are subtracted from the total.
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