Question

Find the center of mass of a thin plate of constant density δ covering the given region. The region bounded by the parabola

x = 2y²-y and the line x = y .
The center of mass is_______ (Type an ordered pair)

Answer 1

Explanation:

o find the center of mass of a thin plate with constant density covering a given region, we need to calculate the coordinates (x, y) of the center of mass using double integrals.

The region is bounded by the parabola x=2y2−yx=2y2−y and the line x=yx=y. To determine the limits of integration, we need to find the points of intersection of these two curves.

Setting x=2y2−yx=2y2−y equal to x=yx=y, we get:

2y2−y=y2y2−y=y

2y2−2y=02y2−2y=0

2y(y−1)=02y(y−1)=0

This gives us two possible solutions: y=0y=0 and y=1y=1.

Now, we'll integrate to find the center of mass using the following formulas:

Center of mass in the x-direction (xcmxcm​):

xcm=1M∬Dxδ dAxcm​=M1​∬D​xδdA

Center of mass in the y-direction (ycmycm​):

ycm=1M∬Dyδ dAycm​=M1​∬D​yδdA

Where MM is the total mass of the plate and DD is the region bounded by the curves.

The density δδ is constant, so it can be factored out of the integrals.

First, let's calculate the total mass MM:

M=∬Dδ dAM=∬D​δdA

Now, calculate the center of mass coordinates:

xcm=1M∬Dxδ dAxcm​=M1​∬D​xδdA

ycm=1M∬Dyδ dAycm​=M1​∬D​yδdA

The ordered pair representing the center of mass will be (xcm,ycm)(xcm​,ycm​).

Please note that the exact calculations depend on the specific limits of integration and the constant density value. Since the calculations can be complex, it's recommended to use appropriate mathematical software or tools for performing the integrals accurately.