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 Statement:**
Sketch the region enclosed by \( y = 7x \) and \( y = 8x^2 \). Decide whether to integrate with respect to \( x \) or \( y \). Then find the area of the region.
**Instructions for Solution:**
1. **Sketching the Region:**
- Begin by graphing the linear equation \( y = 7x \).
- Next, graph the quadratic equation \( y = 8x^2 \).
- Identify the points of intersection by setting the equations equal to each other:
\[ 7x = 8x^2 \]
- Solve for \( x \) to find the points of intersection.
2. **Choosing the Integration Variable:**
- Compare the equations and decide whether integration with respect to \( x \) or \( y \) simplifies calculations.
3. **Calculating the Area:**
- Use the points of intersection to determine the limits of integration.
- Set up the appropriate integral to find the area between the curves.
- Evaluate the integral to find the exact area of the region enclosed.
**Graphical Representation:**
- **Linear Equation \( y = 7x \):** A straight line passing through the origin with a positive slope.
- **Quadratic Equation \( y = 8x^2 \):** A parabola opening upwards with the vertex at the origin.
**Conclusion:**
- Solving the intersection and setting up the integral will provide the area enclosed by the two curves.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8781bdc0-c2b8-434d-8f41-ef58f2d60394%2Fce8fb06a-9823-4e81-83c0-f814b52c2d63%2F70fjelc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Sketch the region enclosed by \( y = 7x \) and \( y = 8x^2 \). Decide whether to integrate with respect to \( x \) or \( y \). Then find the area of the region.
**Instructions for Solution:**
1. **Sketching the Region:**
- Begin by graphing the linear equation \( y = 7x \).
- Next, graph the quadratic equation \( y = 8x^2 \).
- Identify the points of intersection by setting the equations equal to each other:
\[ 7x = 8x^2 \]
- Solve for \( x \) to find the points of intersection.
2. **Choosing the Integration Variable:**
- Compare the equations and decide whether integration with respect to \( x \) or \( y \) simplifies calculations.
3. **Calculating the Area:**
- Use the points of intersection to determine the limits of integration.
- Set up the appropriate integral to find the area between the curves.
- Evaluate the integral to find the exact area of the region enclosed.
**Graphical Representation:**
- **Linear Equation \( y = 7x \):** A straight line passing through the origin with a positive slope.
- **Quadratic Equation \( y = 8x^2 \):** A parabola opening upwards with the vertex at the origin.
**Conclusion:**
- Solving the intersection and setting up the integral will provide the area enclosed by the two curves.
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