Find the center of mass of a thin plate of constant density δcovering the given region. The regi

step 1 Hello, hope you're doing well. So these are our two equations here that we're given. And first w

Find the center of mass of a thin plate of constant density...

Key Concepts

Region of Integration

The region of integration is defined by the area bounded by given curves. Determining the boundaries, such as where two curves intersect, is fundamental for setting the correct limits of integration. A precise description of the region ensures that the calculated area and moments accurately represent the whole plate.

Center of Mass

The center of mass of a plane region is the point at which the weighted relative positions of the distributed mass sum to zero. In the context of a thin plate, it is determined by dividing the moments of the region (which are calculated via integration of the coordinate values times the density) by the total mass. This concept provides a single point representing the balance of the entire mass distribution.

Uniform Density

Uniform density means that the density value is constant throughout the region. This simplifies the computation of the center of mass because the constant density factor cancels out when forming the ratio of the moments to the total mass, allowing the center of mass to be computed as the geometric centroid of the region.

Double Integration

Double integration is used to calculate both the total mass (or area, when density is constant) of the region and its moments about the coordinate axes. By setting up iterated integrals over the region, one can systematically integrate the functions defining the mass distribution and ultimately find the coordinates of the center of mass.

Transcript

00:01 Hello, hope you're doing well.

00:02 So these are our two equations here that we're given.

00:04 And first we want to figure out where these two graphs intersect to figure out the boundaries of our thin plate.

00:11 So since both of these equations are equal to y, we're going to set these two right -hand side of these two equations equal to each other and solve for x.

00:21 So you can add x squared to both sides of our equation and subtract 2x from both sides.

00:30 That gives us 3x squared minus 6x is equal to 0.

00:36 We can factor out of 3x exists with x minus 2 is equal to 0, meaning that x is equal to 0 and x is equal to 2.

00:48 So this is kind of our boundary, x is equal to 0 and x equal to 2.

00:52 These are our limits for our boundary for the thin plate.

00:56 So in order to kind of visualize this, i'm going to draw a quick sketch of what we're looking at.

01:02 So our equations are going to intersect at x is equal to 0 and x is equal to 2.

01:18 So our two equations are going to look just an approximation, something like this.

01:28 And then these are our two intersection points here.

01:32 And this is our region that we want to, that our plate.

01:37 Our thin plate is in.

01:39 So if we defined f of x as the top function and g of x is the bottom function, this is f of x, and this function here is g of x.

01:49 So meaning this f of x function here that forms the top boundary that's going to be our 2x minus x squared, that's f of x.

01:58 And this 2x squared minus 4x, this is going to be our g of x that forms our lower boundary.

02:02 So we'll need this later on in our calculations.

02:06 So first we want to find the mass of our thin plate.

02:11 So mass is defined as the integral from a to b for a density function, which is just a constant in this case, times f of x or top function minus g of x, the bottom function, dx.

02:26 So you can take this delta, this density function out front, then our interval goes from, we got from here that our interval goes from 0 to 2, and our f of x function is 2x minus x squared, and our g of x function is 2x squared minus 4x.

02:52 All right, so you can continue simplifying this.

02:54 This is the integral from 0 to 2 of 2x minus x squared minus 2x squared plus 4x, dx, that's equal to delta times the integral from zero to two.

03:08 Combining like terms, we get 6x minus 3x squared dx.

03:14 And taking the integral of this, that's delta times 6x squared divided by 2, that's 3x squared, minus 3x cubed divided by 3.

03:23 We'll just give us x cubed.

03:24 That goes from 0 to 2.

03:29 That's equal to delta times substituting this upper boundary 2 in.

03:35 3 times 2 squared minus 2 cubed.

03:40 And we don't have to subtract this 0 or plugging in this 0 and subtract that because that'll just give this 0.

03:46 So this is equal to delta times 2 squared is 4 times 3 is 12, minus 2 cubed is 8.

03:54 It's equal to 4 delta, meaning our mass of this thin plate is equal to 4 times delta, or times our density.

04:04 So now we can use this mass to find our, x and y coordinates of our center of mass.

04:09 So our x value for our center of mass is defined as one over the mass times the integral from a to b or delta or density function times x times f of x minus g of x d x.

04:26 So our mass here is equal to four delta so it's equal to 1 over 4 delta can bring out this delta here out front since it's a constant times delta.

04:35 These just cancel out.

04:37 It's the integral from 0 to 2 x times, and we figured out from before what f of x minus g of x is, which just turns out to be 6x minus 3x squared.

04:48 So it's x times 6x minus 3x squared, dx...

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