Circle which value best approximates the definite integral 2 sin nx dx. Provide a convincing 0. argument explaining why you chose that value. Your argument must include a sketch of the graph, or a screenshot of the graph from Desmos. (a) 6 (b) 1/2 (c) 4 (d) 5/4
Circle which value best approximates the definite integral 2 sin nx dx. Provide a convincing 0. argument explaining why you chose that value. Your argument must include a sketch of the graph, or a screenshot of the graph from Desmos. (a) 6 (b) 1/2 (c) 4 (d) 5/4
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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![### Integral Approximation Problem
#### Problem Statement:
Circle which value best approximates the definite integral \( \int_0^{1} 2 \sin \pi x \, dx \). Provide a convincing argument explaining why you chose that value. Your argument must include a sketch of the graph, or a screenshot of the graph from Desmos.
#### Answer Choices:
- (a) 6
- (b) \( \frac{1}{2} \)
- (c) 4
- (d) \( \frac{5}{4} \)
##### Solution Guide:
To solve this problem, follow the steps below:
1. **Understand the Integral:**
- The integral we are asked to approximate is \( \int_0^{1} 2 \sin( \pi x ) \, dx \).
- This represents the area under the curve of the function \( 2 \sin(\pi x) \) from \( x = 0 \) to \( x = 1 \).
2. **Graph the Function:**
- Sketch or use graphing software like Desmos to plot the function \( 2 \sin( \pi x ) \) over the interval \([0, 1]\).
- Note the shape and important points of the sine function, especially at \( x = 0 \) and \( x = 1 \).
3. **Calculate the Integral:**
- Use the fundamental theorem of calculus if comfortable, or approximate the area using trapezoid or Simpson's rules if needed.
- Analytical calculation involves:
\[
\int_0^{1} 2 \sin( \pi x ) \, dx = \left[ -\frac{2}{\pi} \cos(\pi x) \right]_0^1
= -\frac{2}{\pi} \cos(\pi) - \left( -\frac{2}{\pi} \cos(0) \right)
= -\frac{2}{\pi}(-1) + \frac{2}{\pi}(1)
= \frac{4}{\pi}
\]
- Since \(\pi \approx 3.14\), this evaluates to approximately \( \frac{4}{3.14} \approx 1.27 \).
4. **Choose the Closest Value:**
- Compare the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7d6a0751-6896-496c-9850-d9c5dd8d9dc3%2F6b78066a-a8f1-45f6-8523-100013c9eb54%2Fz1yec6ro_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Integral Approximation Problem
#### Problem Statement:
Circle which value best approximates the definite integral \( \int_0^{1} 2 \sin \pi x \, dx \). Provide a convincing argument explaining why you chose that value. Your argument must include a sketch of the graph, or a screenshot of the graph from Desmos.
#### Answer Choices:
- (a) 6
- (b) \( \frac{1}{2} \)
- (c) 4
- (d) \( \frac{5}{4} \)
##### Solution Guide:
To solve this problem, follow the steps below:
1. **Understand the Integral:**
- The integral we are asked to approximate is \( \int_0^{1} 2 \sin( \pi x ) \, dx \).
- This represents the area under the curve of the function \( 2 \sin(\pi x) \) from \( x = 0 \) to \( x = 1 \).
2. **Graph the Function:**
- Sketch or use graphing software like Desmos to plot the function \( 2 \sin( \pi x ) \) over the interval \([0, 1]\).
- Note the shape and important points of the sine function, especially at \( x = 0 \) and \( x = 1 \).
3. **Calculate the Integral:**
- Use the fundamental theorem of calculus if comfortable, or approximate the area using trapezoid or Simpson's rules if needed.
- Analytical calculation involves:
\[
\int_0^{1} 2 \sin( \pi x ) \, dx = \left[ -\frac{2}{\pi} \cos(\pi x) \right]_0^1
= -\frac{2}{\pi} \cos(\pi) - \left( -\frac{2}{\pi} \cos(0) \right)
= -\frac{2}{\pi}(-1) + \frac{2}{\pi}(1)
= \frac{4}{\pi}
\]
- Since \(\pi \approx 3.14\), this evaluates to approximately \( \frac{4}{3.14} \approx 1.27 \).
4. **Choose the Closest Value:**
- Compare the
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