An object is formed so that its base is the quarter circle y = √4x² in the first quadrant, and its cross sections along the x-axis are squares as shown. What is the volume of the object? (Assume the axes are measured in centimeters.) cm3 X y-axis 0 X -√√4-x² 2 -x-axis
An object is formed so that its base is the quarter circle y = √4x² in the first quadrant, and its cross sections along the x-axis are squares as shown. What is the volume of the object? (Assume the axes are measured in centimeters.) cm3 X y-axis 0 X -√√4-x² 2 -x-axis
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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![An object is formed so that its base is the quarter circle \( y = \sqrt{4 - x^2} \) in the first quadrant, and its cross sections along the x-axis are squares as shown. What is the volume of the object? (Assume the axes are measured in centimeters.)
[Input box]
⛌ cm\(^3\)
**Graph Explanation:**
The graph illustrates the geometric figure described in the problem:
- The base of the object is a quarter circle defined by the function \( y = \sqrt{4 - x^2} \) in the first quadrant. This forms a curve from the origin \( (0,0) \) to the point \( (2,0) \) on the x-axis.
- Vertical lines perpendicular to the x-axis extend upwards from the curve. These lines indicate the height of cross-sectional squares.
- The cross-sectional area is a square, the side of which is equal to the y-coordinate at any point along the quarter circle, i.e., the length of the square's side is \( \sqrt{4 - x^2} \).
- The visualization shows the object as a series of these squares stacked along the base curve.
To find the volume, one would typically integrate the area of these squares from \( x = 0 \) to \( x = 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2065d956-6c8d-4dd3-a865-df08e0123eae%2F7c8631bb-3691-4cf8-80c0-f13a98e6c1da%2Fdc9xju_processed.png&w=3840&q=75)
Transcribed Image Text:An object is formed so that its base is the quarter circle \( y = \sqrt{4 - x^2} \) in the first quadrant, and its cross sections along the x-axis are squares as shown. What is the volume of the object? (Assume the axes are measured in centimeters.)
[Input box]
⛌ cm\(^3\)
**Graph Explanation:**
The graph illustrates the geometric figure described in the problem:
- The base of the object is a quarter circle defined by the function \( y = \sqrt{4 - x^2} \) in the first quadrant. This forms a curve from the origin \( (0,0) \) to the point \( (2,0) \) on the x-axis.
- Vertical lines perpendicular to the x-axis extend upwards from the curve. These lines indicate the height of cross-sectional squares.
- The cross-sectional area is a square, the side of which is equal to the y-coordinate at any point along the quarter circle, i.e., the length of the square's side is \( \sqrt{4 - x^2} \).
- The visualization shows the object as a series of these squares stacked along the base curve.
To find the volume, one would typically integrate the area of these squares from \( x = 0 \) to \( x = 2 \).
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