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 an Enclosed Region Using Integration
To find the area of the region enclosed by the equations \( y = 4.75x \) and \( x = 8.5 - y^2 \), follow these steps. The graph provides a visual guide for the enclosed region.
#### Graph Explanation:
The graph displays a region bounded by two curves:
1. \( y = 4.75x \), which is a straight line.
2. \( x = 8.5 - y^2 \), which is a parabola opening to the left.
#### Steps to Find the Enclosed Area:
1. **Use Horizontal Strips to Find the Area**:
Integrate with respect to \( y \) to find the area of the region.
2. **Find the \( y \)-coordinates of Intersection**:
Determine the points where the line \( y = 4.75x \) intersects the parabola \( x = 8.5 - y^2 \).
Solve the system of equations:
\[
y = 4.75x
\]
and
\[
x = 8.5 - y^2
\]
Substitute \( y \) from the first equation into the second equation:
\[
x = 8.5 - (4.75x)^2
\]
Simplify to find the \( y \)-values at the points of intersection. These values will be the limits of integration.
- Lower limit \( c = \) [Enter Value]
- Upper limit \( d = \) [Enter Value]
3. **Set Up the Integral**:
To find the area of the enclosed region from \( c \) to \( d \):
\[
\int_{c}^{d} g(y) \, dy
\]
Here, \( g(y) \) is the difference of the functions representing the curves, rearranged in terms of \( y \) if necessary.
4. **Evaluate the Integral**:
Solve the definite integral to find the area.
- [Set Up Integral]
- Evaluate the definite integral to find the area \( \text{Area} = \) [Enter Value]
### Example Calculation
(For simplicity, assume example values for the coordinates and corresponding integral setup).
1. Determine the intersection points:
- Solve for \( y \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08594285-cc6b-4718-bdd1-7a3127ce35f5%2F6bafa4f3-ebb9-42a9-a593-1c60db41a87f%2Fcaqdrh.png&w=3840&q=75)
Transcribed Image Text:### Finding the Area of an Enclosed Region Using Integration
To find the area of the region enclosed by the equations \( y = 4.75x \) and \( x = 8.5 - y^2 \), follow these steps. The graph provides a visual guide for the enclosed region.
#### Graph Explanation:
The graph displays a region bounded by two curves:
1. \( y = 4.75x \), which is a straight line.
2. \( x = 8.5 - y^2 \), which is a parabola opening to the left.
#### Steps to Find the Enclosed Area:
1. **Use Horizontal Strips to Find the Area**:
Integrate with respect to \( y \) to find the area of the region.
2. **Find the \( y \)-coordinates of Intersection**:
Determine the points where the line \( y = 4.75x \) intersects the parabola \( x = 8.5 - y^2 \).
Solve the system of equations:
\[
y = 4.75x
\]
and
\[
x = 8.5 - y^2
\]
Substitute \( y \) from the first equation into the second equation:
\[
x = 8.5 - (4.75x)^2
\]
Simplify to find the \( y \)-values at the points of intersection. These values will be the limits of integration.
- Lower limit \( c = \) [Enter Value]
- Upper limit \( d = \) [Enter Value]
3. **Set Up the Integral**:
To find the area of the enclosed region from \( c \) to \( d \):
\[
\int_{c}^{d} g(y) \, dy
\]
Here, \( g(y) \) is the difference of the functions representing the curves, rearranged in terms of \( y \) if necessary.
4. **Evaluate the Integral**:
Solve the definite integral to find the area.
- [Set Up Integral]
- Evaluate the definite integral to find the area \( \text{Area} = \) [Enter Value]
### Example Calculation
(For simplicity, assume example values for the coordinates and corresponding integral setup).
1. Determine the intersection points:
- Solve for \( y \
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