The base of a three-dimensional figure is bound by the graph x = y³ and the y-axis on the interval [1, 2]. Vertical cross-sections that are perpendicular to the y-axis are rectangles with height equal to 2. Algebraically, find the area of each rectangle. - 1 2 3 4 5 6 7 8 9 y³ 0 0 O 2y³
The base of a three-dimensional figure is bound by the graph x = y³ and the y-axis on the interval [1, 2]. Vertical cross-sections that are perpendicular to the y-axis are rectangles with height equal to 2. Algebraically, find the area of each rectangle. - 1 2 3 4 5 6 7 8 9 y³ 0 0 O 2y³
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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![### Understanding the Area of Rectangles in Three-Dimensional Figures
**Problem Statement:**
The base of a three-dimensional figure is bounded by the graph \( x = y^3 \) and the y-axis on the interval \([1, 2]\). Vertical cross-sections that are perpendicular to the y-axis are rectangles with a height equal to 2.
**Question:**
Algebraically, find the area of each rectangle.
**Graph and Explanation:**
The provided graph illustrates the curve \( x = y^3 \) and highlights the area of interest between \( y = 1 \) and \( y = 2 \). Here's a detailed breakdown:
- **Axes**: The x-axis and y-axis are labeled and intersect at the origin (0,0).
- **Curve**: The curve \( x = y^3 \) is plotted from \( y = 1 \) to \( y = 2 \).
- **Interval**: The interval on the y-axis is shown from \( y = 1 \) to \( y = 2 \).
- **Shading**: The region under the curve from \( y = 1 \) to \( y = 2 \) is shaded, indicating the base of the three-dimensional figure.
**Calculation:**
The width of each rectangle can be derived from the curve \( x = y^3 \). The area of a rectangle is given by:
\[ \text{Area} = \text{height} \times \text{width} \]
Given height = 2, and the width is \( y^3 \):
\[ \text{Area} = 2y^3 \]
**Available Options:**
- \(\frac{1}{2} y^3\)
- \(2y^3\)
- \(y^3\)
- \(\frac{y^3}{3}\)
**Correct Answer:**
\[ \boxed{2y^3} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf75a45c-687d-4994-8001-f519eebb3c9c%2F77e56c95-0c4a-467a-bb43-0af6ab6da738%2F5v73sye_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Area of Rectangles in Three-Dimensional Figures
**Problem Statement:**
The base of a three-dimensional figure is bounded by the graph \( x = y^3 \) and the y-axis on the interval \([1, 2]\). Vertical cross-sections that are perpendicular to the y-axis are rectangles with a height equal to 2.
**Question:**
Algebraically, find the area of each rectangle.
**Graph and Explanation:**
The provided graph illustrates the curve \( x = y^3 \) and highlights the area of interest between \( y = 1 \) and \( y = 2 \). Here's a detailed breakdown:
- **Axes**: The x-axis and y-axis are labeled and intersect at the origin (0,0).
- **Curve**: The curve \( x = y^3 \) is plotted from \( y = 1 \) to \( y = 2 \).
- **Interval**: The interval on the y-axis is shown from \( y = 1 \) to \( y = 2 \).
- **Shading**: The region under the curve from \( y = 1 \) to \( y = 2 \) is shaded, indicating the base of the three-dimensional figure.
**Calculation:**
The width of each rectangle can be derived from the curve \( x = y^3 \). The area of a rectangle is given by:
\[ \text{Area} = \text{height} \times \text{width} \]
Given height = 2, and the width is \( y^3 \):
\[ \text{Area} = 2y^3 \]
**Available Options:**
- \(\frac{1}{2} y^3\)
- \(2y^3\)
- \(y^3\)
- \(\frac{y^3}{3}\)
**Correct Answer:**
\[ \boxed{2y^3} \]
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