Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.)
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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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