Find the center of mass of a thin plate covering the region bounded below by the parabola y=x^2

Find the center of mass of a thin plate covering the region...

Key Concepts

Center of Mass

The center of mass is the point at which the weighted average of the positions of all mass elements in a body is located. It is found by dividing the first moments (integrals of the coordinate multiplied by density) by the total mass of the object. Calculating it involves integrating over the shape of the region with the given density function.

Density Function

A density function describes how mass is distributed across a region. It can vary with position and is essential in integration for adjusting the weight of different areas according to their mass contributions, thereby accurately capturing the physical characteristics of the object.

Double Integrals

Double integrals are mathematical tools used to compute quantities over a two-dimensional area. They allow for the calculation of total mass and moments by integrating the density function over the region defined by its boundaries, particularly useful when the region is non-standard or bounded by curves.

Moments

Moments are measures that capture the distribution of mass relative to a specific axis. The moments about the coordinate axes are calculated by integrating the product of the density function and the respective coordinate, which are then used to determine the center of mass by balancing these weighted positions.

Transcript

00:01 Okay, so we're asked to find the center of mass at x bar, y bar, of a thin plate that's found by the region y equals x squared and y equals x.

00:21 If the plate's density is given by the function row of x is equal to 12.

00:29 Okay, so if you think about these two curves, they intersect at zero and one.

00:36 So that's x equals 0 and x equals 1.

00:42 The bottom one looks like a parabola and the top one looks like y equals x.

00:52 Okay, so first of all we'll find m is equal to the integral from a to b of our density function times f of x minus g of x.

01:15 Okay, so this is equal to the integral from zero to 1 times 12x times f of x so that's our top function which is right here so that's x minus x squared which is our bottom function okay so this is equal to 12 times the integral from 0 to 1 of distribute this x we have x squared minus x cubed okay so when you evaluate the integral we have 12 times x cubed over 3 minus x to the 4th over 4.

02:12 Evaluated from 0 to 1.

02:14 If i distribute the 12, i have 4 x cubed minus 3 x to 4th, evaluated from 0 to 1.

02:28 When we plug in 1, we get 4 minus 3.

02:31 We plug in zero, we just get zero.

02:34 So our mass is equal to one.

02:41 Next we've got to find a mx.

02:47 So mx is equal to the integral from a to b of row of x over two times f of x squared minus g of x squared.

03:14 Okay, so this is the integral from 0 to 1 times 12x divided by 2 times our top most function is y equals x, so x squared.

03:31 Our bottom most function is y equals x squared, and we square that we get x to the 4th.

03:42 So we can pull 12 divided by 2, which is 6 out front...

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