**Finding the Integrals for the Center of Mass of a Thin Plate:**Given the region bounded below by the parabola \(y = x^2\) and above by the line \(y = x\), and assuming the plate's density is \(\delta(x) = 12x\), we aim to set up the integrals for the center of mass. Do not solve these integrals.---**Formulas:**- **Moment about the x-axis (\( M_x \))**\[ M_x = \int \int y \delta(x) \, dA \]- **Moment about the y-axis (\( M_y \))**\[ M_y = \int \int x \delta(x) \, dA \]- **Mass (\( M \))**\[ M = \int \int \delta(x) \, dA \]---**Integrals Setup:**1. **Moment about the x-axis (\( M_x \))** \[ M_x = \] \[ \int_{a}^{b} \int_{g(x)}^{f(x)} y \delta(x) \, dy \, dx \]2. **Moment about the y-axis (\( M_y \))** \[ M_y = \] \[ \int_{a}^{b} \int_{g(x)}^{f(x)} x \delta(x) \, dy \, dx \]3. **Mass (\( M \))** \[ M = \] \[ \int_{a}^{b} \int_{g(x)}^{f(x)} \delta(x) \, dy \, dx \]---**Details of the Region:**1. The parabola is represented by \( y = x^2 \).2. The line is represented by \( y = x \).3. The density function is given by \( \delta(x) = 12x \).4. The region of integration is defined by the intersection points of \( y = x^2 \) and \( y = x \). Calculating these intersection points: \[ x^2 = x \implies x(x - 1) = 0 \implies x = 0 \text{ or } x = 1 \]Therefore

We have to dind the integrals for the center of mass of a thin plate covering the region bounded…

Answered: Find the integrals for the center of…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.CR: Chapter 9 Review

Problem 54CR

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Find the integrals for the center of mass of a thin plate covering the region bounded below by the parabola y = x² and above by the line y = x if the plate's density is 8(x) = 122. SET UP THE INTEGRALS ONLY!!! DO NOT SOLVE!!! M₂ = My= 1 JH M = dx Preview da Preview