Find the area of the region. Use a graphing utility to verify your result. -π/4 Jπ/12 4 3 2 1 csc(2x) cot(2x) dx 16 1100 8 Зл 16 T X ℗
Find the area of the region. Use a graphing utility to verify your result. -π/4 Jπ/12 4 3 2 1 csc(2x) cot(2x) dx 16 1100 8 Зл 16 T X ℗
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Finding the Area of a Region**
To find the area of the region, use a graphing utility to verify your result. The task involves evaluating the integral:
\[
\int_{\pi/12}^{\pi/4} \text{csc}(2x) \cdot \text{cot}(2x) \, dx
\]
**Graph Analysis**
The graph depicts the curve of the function \( y = \text{csc}(2x) \cdot \text{cot}(2x) \) within the specified limits \(\pi/12\) to \(\pi/4\), plotted along the x-axis.
- The x-axis is marked with interval notations: \( \frac{\pi}{16} \), \( \frac{\pi}{8} \), \( \frac{3\pi}{16} \), and \( \frac{\pi}{4} \).
- The y-axis is labeled with integer values from 1 to 4, representing the function's output.
- A shaded region indicates the area under the curve that corresponds to the evaluated integral.
**Additional Resources**
- Buttons labeled "Read It" and "Master It" are present to provide additional help and mastery of the content.
This visualization aids in understanding how the definite integral corresponds to the area under a curve within specific limits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62219f22-1a6a-4c46-9e25-1c8e624a2ed8%2Ff79b5dd1-7b68-4de1-b46c-dd865a2f98c2%2Ficzgg1v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Area of a Region**
To find the area of the region, use a graphing utility to verify your result. The task involves evaluating the integral:
\[
\int_{\pi/12}^{\pi/4} \text{csc}(2x) \cdot \text{cot}(2x) \, dx
\]
**Graph Analysis**
The graph depicts the curve of the function \( y = \text{csc}(2x) \cdot \text{cot}(2x) \) within the specified limits \(\pi/12\) to \(\pi/4\), plotted along the x-axis.
- The x-axis is marked with interval notations: \( \frac{\pi}{16} \), \( \frac{\pi}{8} \), \( \frac{3\pi}{16} \), and \( \frac{\pi}{4} \).
- The y-axis is labeled with integer values from 1 to 4, representing the function's output.
- A shaded region indicates the area under the curve that corresponds to the evaluated integral.
**Additional Resources**
- Buttons labeled "Read It" and "Master It" are present to provide additional help and mastery of the content.
This visualization aids in understanding how the definite integral corresponds to the area under a curve within specific limits.
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