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 first we want to figure out what the region bounded by the gra

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

Key Concepts

Center of Mass

The center of mass of a region is the ‘balance point’ where the weighted average of all the points in the region (with respect to density) is computed. In the context of a thin plate, its coordinates are found by dividing the first moment (integral of the coordinate times area element) by the total mass (or area, when density is constant), yielding x? and ?.

Constant Density and Thin Plate Concepts

When dealing with a thin plate of constant density, the constant can be factored out in the integrals for both the total mass and the moments, simplifying the computation. This principle allows the calculation to often reduce to finding area and moments of the geometric region without concern for variable density influences.

Double Integration for Area and Moments

Double integration is used to compute both the total area (or mass) and the moments (integrals of x and y over the region). The general formulas for the center of mass involve setting up integrals over the region to sum the contributions of infinitesimal elements, which are then normalized by the total area or mass.

Setting Up the Region of Integration

Defining the bounds of integration correctly is crucial when the region is described by curves. This process involves identifying the intersection points of the boundaries and determining appropriate limits for each variable, whether the integration is done with respect to x first or y. For example, when the region is bounded by a curve and a line, one must determine the x-values where these curves intersect and then express the y-limits in terms of x, or vice versa.

Transcript

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

00:02 So first we want to figure out what the region bounded by the graph of these two equations is.

00:08 So you can do a quick sketch.

00:16 So y is equal to x squared it's going to have points right there 1 -1 and minus 1 1 1 in 2 4 and minus 2 4.

00:29 So this is going to be the graph of our y is equal to x squared function and then y is equal to 4 that's just going to be a horizontal line that y is equal to 4.

00:38 So their intersection points are at x is equal to minus 2, and x is equal to 2.

00:45 So this is going to be our range.

00:47 We're trying to find the plate that covers this area in here.

00:50 So the range of our x values is going to be minus 2 to 2.

00:54 So first we need to find out what the mass of this plate is.

01:00 So our mass is going to be equal to an integral, an a to b, of our density function times f of x minus g of x d x so move this down a little bit our range like we figured out before is going from minus two to two is going to be from minus two to two our density function is just constant delta times our f of x function is our top function which is four minus our bottom function which is x squared dx.

01:40 We can take out this delta here, this density function that goes from minus 2 to 2, 4 minus x squared dx.

01:50 And then now we can take this integral.

01:53 So we've got our delta times our integral of 4 is going to minus x squared is going to minus x cubed divided by 3.

02:04 So it's minus 1 3rd x cube, going from minus 2 to 2 to 2.

02:07 It's going to be our density.

02:09 Function, i'll move that down to the next line, it's going to be equal to our density function times four times, so first we're going to plug in two, just go four times two minus one -third times two cubed, minus, now we're going to plug in this minus two to four times minus two, minus one -third times minus two cubed.

02:37 All right, so now simplifying this, we have delta, we have four times two is eight, minus two cubed is eight divided by three.

02:47 So it's going to eight minus eight thirds.

02:50 Four times minus two is minus eight.

02:54 So minus two cubes is going to give us minus eight times minus one thirds is going to be plus eight thirds.

03:04 Simplifying this, we've got, again, our delta, our density times eight.

03:10 Make that a common, have, if we convert everything to a common denominator of three, we're going to have 24 thirds minus 8 thirds plus 8 or it's going to be plus 24 thirds minus 8 thirds.

03:31 So it's going to be 24 plus 24 is 48 minus 16.

03:37 That's going to give us 32 thirds.

03:43 So which means our mass value is equal to 32 times our depth.

03:48 Density over 3.

03:51 All right.

03:51 So we got our mass value.

03:53 So now we want to figure out our centroid or our center of mass.

03:59 So to do that, our x value for our center of mass is going to be 1 over m.

04:06 The integral of delta x is going to be a to b times f of x minus g of x d x.

04:19 So we can take so first we know that our mass is 32 delta over 3, or 32 times of density over 3.

04:26 So 1 over mass is going to be 3 over 32 times our density function.

04:34 We can take out this density function here in front.

04:38 These just cancel out and go to 0.

04:40 So we're left with the integral from minus 2 to 2 of x times our f of x is 4 minus our g of x, which is x squared.

04:50 It's dx.

04:51 So now taking this integral, this is, or we need to simplify this a little bit more, 3 over 32, minus 2 to 2, x times 4 is 4x, x times minus x squared is minus x cubed, dx.

05:08 Now if we take this integral, it gives us 3 over 32 times integral of 4x, that's going to be 4x squared over 2, so it's going to be 2x squared, minus one -fourth x to the fourth.

05:24 We're going to go from minus 2 to 2.

05:27 Now simplifying this, that's equal to 3 over 32, times first plugging in this 2 up here, our upper boundary, we have 2 times 2 squared, minus 1 4th times 2 to the 4th.

05:44 Now we're going to subtract, plugging in our lower boundary here, minus 2.

05:49 It's going to be 2 times minus 2 squared.

05:52 Minus one -fourth times minus two to the fourth.

05:59 And now we just want to simplify this.

06:02 We've got 3 over 32 times 2 times 2 times 2 square...

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