Find the area of the surface generated when the curve: =y=+on [1,2] is rotated about the x-axis. 2 0 2
Find the area of the surface generated when the curve: =y=+on [1,2] is rotated about the x-axis. 2 0 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Find the area of the surface generated when the curve:
a) y=(x^4)/8+1/(4x^2) on [1,2] is rotated about the x-axis.
b) x=2sqrt(4-y), 0=<y=<15/4 is rotated about the y-axis.
![**Problem 4: Surface Area Generated by Rotating Curves**
**Objective:** Find the area of the surface generated when the given curves are rotated about the specified axes.
**a) Curve:**
\[ y = \frac{x^4}{8} + \frac{1}{4x^2} \quad \text{on} \quad [1, 2] \]
**Axis of Rotation:** x-axis
**Description:** The graph shows a curve with the specified equation on the domain from x = 1 to x = 2. As the curve rises, it follows a smooth path typical of polynomial and rational function characteristics, displaying an increase in the y-values.
**b) Curve:**
\[ x = 2\sqrt{4 - y}, \quad 0 \leq y \leq \frac{15}{4} \]
**Axis of Rotation:** y-axis
**Explanation:**
- The expression represents a curve described by x in terms of y, indicating that the graph is a portion of a function rotated about the y-axis.
- The limits for y range from 0 to \(\frac{15}{4}\), dictating the portion of the graph under consideration for rotation.
**Instruction:** To find the surface area generated by these rotations, one would typically use integral calculus, specifically the surface area of revolution formula. Each problem involves calculating the definite integral that accounts for the respective curve's revolution around the given axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcad422bc-64ba-4541-be23-0cacc94d0349%2Fc6c81929-6e40-40fc-ba21-760cccf68ada%2Fk00uax_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4: Surface Area Generated by Rotating Curves**
**Objective:** Find the area of the surface generated when the given curves are rotated about the specified axes.
**a) Curve:**
\[ y = \frac{x^4}{8} + \frac{1}{4x^2} \quad \text{on} \quad [1, 2] \]
**Axis of Rotation:** x-axis
**Description:** The graph shows a curve with the specified equation on the domain from x = 1 to x = 2. As the curve rises, it follows a smooth path typical of polynomial and rational function characteristics, displaying an increase in the y-values.
**b) Curve:**
\[ x = 2\sqrt{4 - y}, \quad 0 \leq y \leq \frac{15}{4} \]
**Axis of Rotation:** y-axis
**Explanation:**
- The expression represents a curve described by x in terms of y, indicating that the graph is a portion of a function rotated about the y-axis.
- The limits for y range from 0 to \(\frac{15}{4}\), dictating the portion of the graph under consideration for rotation.
**Instruction:** To find the surface area generated by these rotations, one would typically use integral calculus, specifically the surface area of revolution formula. Each problem involves calculating the definite integral that accounts for the respective curve's revolution around the given axis.
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