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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![### Finding the Area of a Bounded Region
#### Problem Statement:
Find the area of the region bounded to the right by \( x = y + 2 \) and to the left by \( x = y^2 \).
#### Explanation:
The problem requires us to determine the area enclosed between two curves:
1. \( x = y + 2 \)
2. \( x = y^2 \)
#### Graphical Representation:
The image features a graph with two curves:
- The first curve is a straight line represented by \( x = y + 2 \).
- The second curve is a parabola represented by \( x = y^2 \).
Both curves intersect, forming a bounded region whose area needs to be calculated.
#### Graph Details:
- **Axes**:
- The x-axis and y-axis are both labeled and marked with intervals.
- Both axes range from \(-5\) to \(5\).
- **Curves**:
- The line \( x = y + 2 \) starts from the point \((-5, -7)\) and progresses linearly through \( (0, -2) \), intersecting the y-axis at \( y = -2 \).
- The parabola \( x = y^2 \) is upward opening and symmetric about the y-axis. It intersects the x-axis at \((0, 0)\) and \((4, 2)\).
#### Area Calculation:
To find the area of the region bounded by these two curves, set them equal to find points of intersection:
\[ y + 2 = y^2 \]
\[ y^2 - y - 2 = 0 \]
\[ (y - 2)(y + 1) = 0 \]
Thus, \( y = 2 \) and \( y = -1 \).
The bounded region is from \( y = -1 \) to \( y = 2 \).
Then, the area \( A \) can be calculated by integrating the difference between the line and the parabola within these limits:
\[ A = \int_{-1}^{2} [(y + 2) - y^2] \, dy \]
1. Evaluate the integral:
\[ A = \int_{-1}^{2} (y + 2 - y^2) \, dy \]
\[ = \left[ \frac{y^2}{2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bc5b610-4c42-40dc-b81a-6ff9e1fb2857%2F5a80d3eb-6a69-4d9e-92eb-a36eec42273a%2F17l39vr_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Area of a Bounded Region
#### Problem Statement:
Find the area of the region bounded to the right by \( x = y + 2 \) and to the left by \( x = y^2 \).
#### Explanation:
The problem requires us to determine the area enclosed between two curves:
1. \( x = y + 2 \)
2. \( x = y^2 \)
#### Graphical Representation:
The image features a graph with two curves:
- The first curve is a straight line represented by \( x = y + 2 \).
- The second curve is a parabola represented by \( x = y^2 \).
Both curves intersect, forming a bounded region whose area needs to be calculated.
#### Graph Details:
- **Axes**:
- The x-axis and y-axis are both labeled and marked with intervals.
- Both axes range from \(-5\) to \(5\).
- **Curves**:
- The line \( x = y + 2 \) starts from the point \((-5, -7)\) and progresses linearly through \( (0, -2) \), intersecting the y-axis at \( y = -2 \).
- The parabola \( x = y^2 \) is upward opening and symmetric about the y-axis. It intersects the x-axis at \((0, 0)\) and \((4, 2)\).
#### Area Calculation:
To find the area of the region bounded by these two curves, set them equal to find points of intersection:
\[ y + 2 = y^2 \]
\[ y^2 - y - 2 = 0 \]
\[ (y - 2)(y + 1) = 0 \]
Thus, \( y = 2 \) and \( y = -1 \).
The bounded region is from \( y = -1 \) to \( y = 2 \).
Then, the area \( A \) can be calculated by integrating the difference between the line and the parabola within these limits:
\[ A = \int_{-1}^{2} [(y + 2) - y^2] \, dy \]
1. Evaluate the integral:
\[ A = \int_{-1}^{2} (y + 2 - y^2) \, dy \]
\[ = \left[ \frac{y^2}{2
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