Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y = √9-x², -2≤x≤2 2x dx = Find M My and (x, y) for the lamina of uniform density p bounded by the graphs of the equations. y=x2/3 Y (x, y) =
Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y = √9-x², -2≤x≤2 2x dx = Find M My and (x, y) for the lamina of uniform density p bounded by the graphs of the equations. y=x2/3 Y (x, y) =
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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![### Calculus Problem Set: Surface Area and Center of Mass
#### Problem 1: Definite Integral for Surface Area
**Task:**
Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis.
Given:
\[ y = \sqrt{9 - x^2}, \quad -2 \leq x \leq 2 \]
Integral Format:
\[ 2\pi \int_{-2}^{2} (\text{Function}) \, dx = \text{Result} \]
#### Problem 2: Moments and Center of Mass
**Task:**
Find \( M_x \), \( M_y \), and \( (\overline{x}, \overline{y}) \) for the lamina of uniform density \( \rho \) bounded by the graphs of the equations.
Given:
\[ y = x^{2/3}, \quad y = \frac{1}{2}x \]
* Moments \( M_x \) and \( M_y \):
\[ M_x = \boxed{\phantom{\int\int}} \]
\[ M_y = \boxed{\phantom{\int\int}} \]
* Center of Mass \( (\overline{x}, \overline{y}) \):
\[ (\overline{x}, \overline{y}) = \boxed{\phantom{(x, y)}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fdef185-de8b-42f5-933e-8018dcf36541%2F317425b0-8b74-4278-afc5-1824a9f805f1%2Fxcmhi7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem Set: Surface Area and Center of Mass
#### Problem 1: Definite Integral for Surface Area
**Task:**
Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis.
Given:
\[ y = \sqrt{9 - x^2}, \quad -2 \leq x \leq 2 \]
Integral Format:
\[ 2\pi \int_{-2}^{2} (\text{Function}) \, dx = \text{Result} \]
#### Problem 2: Moments and Center of Mass
**Task:**
Find \( M_x \), \( M_y \), and \( (\overline{x}, \overline{y}) \) for the lamina of uniform density \( \rho \) bounded by the graphs of the equations.
Given:
\[ y = x^{2/3}, \quad y = \frac{1}{2}x \]
* Moments \( M_x \) and \( M_y \):
\[ M_x = \boxed{\phantom{\int\int}} \]
\[ M_y = \boxed{\phantom{\int\int}} \]
* Center of Mass \( (\overline{x}, \overline{y}) \):
\[ (\overline{x}, \overline{y}) = \boxed{\phantom{(x, y)}} \]
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