Find the volume formed by rotating the region enclosed by: y = 3√ and y = x about the line = 10 243 2 =
Find the volume formed by rotating the region enclosed by: y = 3√ and y = x about the line = 10 243 2 =
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
![**Volume of Rotated Region**
**Problem Statement:**
Find the volume formed by rotating the region enclosed by the curves \( y = 3\sqrt{x} \) and \( y = x \) about the line \( x = 10 \).
**Solution:**
The volume is given by:
\[ \frac{243\pi}{2} \]
**Explanation:**
To solve this problem, one typically uses the method of cylindrical shells or the washer method, depending on the setup of the problem. The graphs \( y = 3\sqrt{x} \) and \( y = x \) intersect at the points where these equations are equal. The region between these curves from the x-axis is then rotated about the vertical line \( x = 10 \), creating a three-dimensional solid. Calculations involve setting up an integral to find this volume, considering the distance from the axis of rotation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffebd70f6-1305-45b0-937a-417a1969a2a1%2F3eeea922-a41b-4569-b7c3-6b0426ef9ef1%2Fjnuzlaq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Volume of Rotated Region**
**Problem Statement:**
Find the volume formed by rotating the region enclosed by the curves \( y = 3\sqrt{x} \) and \( y = x \) about the line \( x = 10 \).
**Solution:**
The volume is given by:
\[ \frac{243\pi}{2} \]
**Explanation:**
To solve this problem, one typically uses the method of cylindrical shells or the washer method, depending on the setup of the problem. The graphs \( y = 3\sqrt{x} \) and \( y = x \) intersect at the points where these equations are equal. The region between these curves from the x-axis is then rotated about the vertical line \( x = 10 \), creating a three-dimensional solid. Calculations involve setting up an integral to find this volume, considering the distance from the axis of rotation.
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