Find the volume of the solid obtained by rotating the region bounded by y = 9x2, x = 4, x = 5, and y = 0, about the x-axis. V =
Find the volume of the solid obtained by rotating the region bounded by y = 9x2, x = 4, x = 5, and y = 0, about the x-axis. V =
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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![**Problem: Volume of a Solid of Revolution**
Find the volume of the solid obtained by rotating the region bounded by \( y = 9x^2 \), \( x = 4 \), \( x = 5 \), and \( y = 0 \), about the \( x \)-axis.
\[ V = \, \boxed{\phantom{V = }} \]
**Explanation:**
To solve this problem using the disk method, integrate the area of circular disks along the \( x \)-axis. The volume \( V \) is given by:
\[ V = \int_{a}^{b} \pi [f(x)]^2 \, dx \]
Where:
- \( f(x) = 9x^2 \) is the radius of the disks,
- \( a = 4 \) and \( b = 5 \) are the bounds of integration.
The integral becomes:
\[ V = \int_{4}^{5} \pi (9x^2)^2 \, dx = \int_{4}^{5} \pi (81x^4) \, dx \]
Calculating the integral will give the volume of the solid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e916346-eb1f-468f-a417-f3db3524f0d2%2Fc8bdc1b3-400a-4c70-b2db-930fcae8aade%2Fwznlcs6_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Volume of a Solid of Revolution**
Find the volume of the solid obtained by rotating the region bounded by \( y = 9x^2 \), \( x = 4 \), \( x = 5 \), and \( y = 0 \), about the \( x \)-axis.
\[ V = \, \boxed{\phantom{V = }} \]
**Explanation:**
To solve this problem using the disk method, integrate the area of circular disks along the \( x \)-axis. The volume \( V \) is given by:
\[ V = \int_{a}^{b} \pi [f(x)]^2 \, dx \]
Where:
- \( f(x) = 9x^2 \) is the radius of the disks,
- \( a = 4 \) and \( b = 5 \) are the bounds of integration.
The integral becomes:
\[ V = \int_{4}^{5} \pi (9x^2)^2 \, dx = \int_{4}^{5} \pi (81x^4) \, dx \]
Calculating the integral will give the volume of the solid.
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