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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Question
![**Problem Statement:**
Find the area between the curves \( y = 2 \sin x \), \( y = \sin 2x \), and the lines \( x = 0 \) and \( x = \pi \). Sketch the region.
**Explanation:**
To solve this problem, you need to calculate the area enclosed by the two given curves over the specified interval from \( x = 0 \) to \( x = \pi \).
**Steps:**
1. **Understand the Functions:**
- \( y = 2 \sin x \) is a sine wave with an amplitude of 2.
- \( y = \sin 2x \) is a sine wave with double the frequency.
2. **Identify Points of Intersection:**
- Set the equations equal to find intersection points: \( 2 \sin x = \sin 2x \).
- Solve for \( x \) within the interval [0, \( \pi \)] to find critical points.
3. **Calculate the Area:**
- Set up the integral of the difference between the two functions over the interval from 0 to \( \pi \) where one function is above the other.
- Evaluate the integral to find the enclosed area.
4. **Sketch the Region:**
- Plot both functions on the same graph.
- Shade the region between the curves from \( x = 0 \) to \( x = \pi \).
By following these steps, you will determine the area between the two curves within the given range.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe73d2f7-f48d-4799-ad26-f906d4f73fa5%2Fe0254bb1-6c9d-4dab-9146-6ad09ad51f0f%2F6hhu6e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the area between the curves \( y = 2 \sin x \), \( y = \sin 2x \), and the lines \( x = 0 \) and \( x = \pi \). Sketch the region.
**Explanation:**
To solve this problem, you need to calculate the area enclosed by the two given curves over the specified interval from \( x = 0 \) to \( x = \pi \).
**Steps:**
1. **Understand the Functions:**
- \( y = 2 \sin x \) is a sine wave with an amplitude of 2.
- \( y = \sin 2x \) is a sine wave with double the frequency.
2. **Identify Points of Intersection:**
- Set the equations equal to find intersection points: \( 2 \sin x = \sin 2x \).
- Solve for \( x \) within the interval [0, \( \pi \)] to find critical points.
3. **Calculate the Area:**
- Set up the integral of the difference between the two functions over the interval from 0 to \( \pi \) where one function is above the other.
- Evaluate the integral to find the enclosed area.
4. **Sketch the Region:**
- Plot both functions on the same graph.
- Shade the region between the curves from \( x = 0 \) to \( x = \pi \).
By following these steps, you will determine the area between the two curves within the given range.
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