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

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Chapter1: Functions And Models
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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
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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