The rectangle shown below has vertices of open parentheses 0 comma space O close parentheses comma space open parentheses straight pi over 2 comma space O close 0 parentheses comma space open parentheses straight pi over 2 comma space 1 close parentheses comma space and space open parentheses 0 comma space 1 close parentheses: What percent of the rectangle is shaded? 36.3% 63.7% 43.2% y=cos(x) 31.4%
The rectangle shown below has vertices of open parentheses 0 comma space O close parentheses comma space open parentheses straight pi over 2 comma space O close 0 parentheses comma space open parentheses straight pi over 2 comma space 1 close parentheses comma space and space open parentheses 0 comma space 1 close parentheses: What percent of the rectangle is shaded? 36.3% 63.7% 43.2% y=cos(x) 31.4%
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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Question
The shaded area is bounded below by the curve \( y = \cos(x) \).
**Visual Description:**
- The \( x \)-axis is labeled.
- The \( y \)-axis is labeled.
- The curve \( y = \cos(x) \) is drawn from \( x = 0 \) to \( x = \frac{\pi}{2} \).
- The rectangle is shaded below the curve \( y = \cos(x) \).
**Question:**
What percent of the rectangle is shaded?
- \( 36.3\% \)
- \( 63.7\% \)
- \( 43.2\% \)
- \( 31.4\% \)
**Explanation:**
The correct answer is highlighted as \( 63.7\% \).
This involves calculating the area under the curve \( y = \cos(x) \) from \( x = 0 \) to \( x = \frac{\pi}{2} \), then expressing this area as a percentage of the area of the rectangle.
To calculate the shaded area:
1. **Rectangle Area**: \( \text{Area}_{\text{rectangle}} = \text{Width} \times \text{Height} = \frac{\pi}{2} \times 1 = \frac{\pi}{2} \)
2. **Area under the curve} \( y = \cos(x) \) from \( 0 \) to \( \frac{\pi}{2} \):
\[
\text{Area}_{\text{curve}} = \int_{0}^{\frac{\pi}{2}} \cos(x) \, dx = \sin(x) \Big|_0^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1
\]
3. **Shaded Area**: \( \text{Shaded Area} = \text{Area}_{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bed085c-3ecb-47a6-9147-a081ba32700e%2F22d734a1-8475-4d59-ad65-80cce1e4039a%2Fpmh9fhr_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The rectangle shown below has vertices at \( (0, 0) \), \( \left(\frac{\pi}{2}, 0\right) \), \( \left(\frac{\pi}{2}, 1\right) \), and \( (0, 1) \):
The shaded area is bounded below by the curve \( y = \cos(x) \).
**Visual Description:**
- The \( x \)-axis is labeled.
- The \( y \)-axis is labeled.
- The curve \( y = \cos(x) \) is drawn from \( x = 0 \) to \( x = \frac{\pi}{2} \).
- The rectangle is shaded below the curve \( y = \cos(x) \).
**Question:**
What percent of the rectangle is shaded?
- \( 36.3\% \)
- \( 63.7\% \)
- \( 43.2\% \)
- \( 31.4\% \)
**Explanation:**
The correct answer is highlighted as \( 63.7\% \).
This involves calculating the area under the curve \( y = \cos(x) \) from \( x = 0 \) to \( x = \frac{\pi}{2} \), then expressing this area as a percentage of the area of the rectangle.
To calculate the shaded area:
1. **Rectangle Area**: \( \text{Area}_{\text{rectangle}} = \text{Width} \times \text{Height} = \frac{\pi}{2} \times 1 = \frac{\pi}{2} \)
2. **Area under the curve} \( y = \cos(x) \) from \( 0 \) to \( \frac{\pi}{2} \):
\[
\text{Area}_{\text{curve}} = \int_{0}^{\frac{\pi}{2}} \cos(x) \, dx = \sin(x) \Big|_0^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1
\]
3. **Shaded Area**: \( \text{Shaded Area} = \text{Area}_{\
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