Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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 \( (\overline{x}, \overline{y}) \): \[ (\overline{x}, \overline{y}) = \boxed{\phantom{(x, y)}} \]
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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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