Let the infinite region bounded by y I-axis. The volume of the infinite solid will be T8 3 the x-axis, and to the right of x = 7 be revolved around the
Let the infinite region bounded by y I-axis. The volume of the infinite solid will be T8 3 the x-axis, and to the right of x = 7 be revolved around the
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
Q12
![**Infinite Solid Volume Calculation**
Consider the infinite region bounded by the function \( y = \frac{1}{x^8} \), the x-axis, and the vertical line \( x = 7 \). This region is to be revolved around the x-axis to form a solid.
The task is to find the volume of the infinite solid thus created.
**Description of Diagrams or Graphs:**
In this scenario, there is no graph or diagram provided. However, if there were, it would typically show:
- The curve of the function \( y = \frac{1}{x^8} \), which decreases steeply for \( x > 7 \).
- The area under the curve starting from \( x = 7 \) and extending to infinity.
- The x-axis as the axis of revolution.
**Volume Calculation:**
To find the volume of the solid formed, use the method of disks or washers, integrating from the boundary at \( x = 7 \) to infinity. The formula to compute the volume \( V \) is:
\[ V = \pi \int_7^\infty \left(\frac{1}{x^8}\right)^2 \, dx \]
(Note: The solution box is provided in the original context for filling in the calculated volume.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c5164fa-e600-4317-9bd4-8c706ea2635f%2Fe01352aa-77e4-4e52-af7e-f249d091b880%2Fl7670qa_processed.png&w=3840&q=75)
Transcribed Image Text:**Infinite Solid Volume Calculation**
Consider the infinite region bounded by the function \( y = \frac{1}{x^8} \), the x-axis, and the vertical line \( x = 7 \). This region is to be revolved around the x-axis to form a solid.
The task is to find the volume of the infinite solid thus created.
**Description of Diagrams or Graphs:**
In this scenario, there is no graph or diagram provided. However, if there were, it would typically show:
- The curve of the function \( y = \frac{1}{x^8} \), which decreases steeply for \( x > 7 \).
- The area under the curve starting from \( x = 7 \) and extending to infinity.
- The x-axis as the axis of revolution.
**Volume Calculation:**
To find the volume of the solid formed, use the method of disks or washers, integrating from the boundary at \( x = 7 \) to infinity. The formula to compute the volume \( V \) is:
\[ V = \pi \int_7^\infty \left(\frac{1}{x^8}\right)^2 \, dx \]
(Note: The solution box is provided in the original context for filling in the calculated volume.)
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