A region bounded by fly) = volume of the solid formed 9 8 7 6 54 4 31 NO 2- -2-11 1 2 3 4 5 6 7 8 X y, y = 1, y = 8, and x = 0 is shown below. Find the by revolving the region about the y-axis.
A region bounded by fly) = volume of the solid formed 9 8 7 6 54 4 31 NO 2- -2-11 1 2 3 4 5 6 7 8 X y, y = 1, y = 8, and x = 0 is shown below. Find the by revolving the region about the y-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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Question
![**Problem Statement:**
A region bounded by \( f(y) = \sqrt[3]{y} \), \( y = 1 \), \( y = 8 \), and \( x = 0 \) is shown below. Find the volume of the solid formed by revolving the region about the y-axis.
**Explanation of Diagram:**
The diagram shows a graph with the x and y axes. The region of interest is shaded and lies between the curve \( f(y) = \sqrt[3]{y} \) and the y-axis from \( y = 1 \) to \( y = 8 \).
- The x-axis ranges from -2 to 8.
- The y-axis ranges from 1 to 9.
- The curve \( f(y) = \sqrt[3]{y} \) is shown as a green line moving from point (1, 1) up to approximately (4, 9).
- The region is shaded with diagonal lines and is bounded on the left by \( x = 0 \), which is the y-axis, and on the right by the curve. The horizontal boundaries are \( y = 1 \) and \( y = 8 \).
**Objective:**
Calculate the volume of the solid when the shaded region is revolved around the y-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa18a6723-a2ab-43ae-9c9a-35233fc8cb89%2Fa808f2ff-5b9d-46e8-b72c-d1644509286c%2Fgm9xnug_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A region bounded by \( f(y) = \sqrt[3]{y} \), \( y = 1 \), \( y = 8 \), and \( x = 0 \) is shown below. Find the volume of the solid formed by revolving the region about the y-axis.
**Explanation of Diagram:**
The diagram shows a graph with the x and y axes. The region of interest is shaded and lies between the curve \( f(y) = \sqrt[3]{y} \) and the y-axis from \( y = 1 \) to \( y = 8 \).
- The x-axis ranges from -2 to 8.
- The y-axis ranges from 1 to 9.
- The curve \( f(y) = \sqrt[3]{y} \) is shown as a green line moving from point (1, 1) up to approximately (4, 9).
- The region is shaded with diagonal lines and is bounded on the left by \( x = 0 \), which is the y-axis, and on the right by the curve. The horizontal boundaries are \( y = 1 \) and \( y = 8 \).
**Objective:**
Calculate the volume of the solid when the shaded region is revolved around the y-axis.
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