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For the following exercises, compute the center of mass $(\overline{x}, \overline{y}) .$ Use symmetry to help locate the center of mass whenever possible.$\rho=7$ in the square $0 \leq x \leq 1, \quad 0 \leq y \leq 1$

$\left(\frac{1}{2}, \frac{1}{2}\right)$

Calculus 1 / AB

Calculus 2 / BC

Chapter 2

Applications of Integration

Section 6

Moments and Centers of Mass

Integrals

Integration

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

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in the following problem. We want to compute the center of mass which is given by these terms right here with information that the density function. Ricky recycle to seven on the interval from Syria to one for X and on the interval. For what? From why sickle to Syria, to a white school to one. So let's first for a picture of how this looks like. So we have over here Xs equal to one and why every six equal to one. So we're looking at these points right here. So this is a rick check. Sorry. A square with Britta's is at one comma one zero comma one zero comma zero no one comma zero. So for the center of master will actually need to use integral Sze However, we can skip that process by looking at this shape right here We know that this is square right here has symmetry are prone to lines The line X is equal to 1/2 and the line why sickle to 1/2 Because this is square has symmetry around these two lines and because the density is a constant function, we can ensure that the center of a mess it's gonna be looking at this point right here in the middle, in the intersection of this to excess of symmetry, which is gonna be at the point 1/2 coma, 1/2 using symmetry, we actually skip the process of using integral to find the center of mess.

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