Question
Asked Oct 31, 2019

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.

D is the triangular region with vertices (0, 0), (2, 1), (0, 3); ρ(x, y) = 8(x + y)
check_circleExpert Solution
Step 1

Given:

The triangular region D with uniform density p(x, y) 8 (x+ y) has vertices
(0,0),(2,1) and (0,3)
From the points (0,0).(2,1),we have
1-0
y-1 (x-.
2 0
2)
2у -2 %3Dх -2
2у %3D х
X
y=-
2
help_outline

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The triangular region D with uniform density p(x, y) 8 (x+ y) has vertices (0,0),(2,1) and (0,3) From the points (0,0).(2,1),we have 1-0 y-1 (x-. 2 0 2) 2у -2 %3Dх -2 2у %3D х X y=- 2

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Step 2

Also,

Another equation is obtained from the points (2,1),(0,3),we have
3-1
(x-0)
y-3=
0-2
(0-)
y-3-x
y=-x+3
to -x3 andx varies from 0 to 2
2
That is, y varies from
help_outline

Image Transcriptionclose

Another equation is obtained from the points (2,1),(0,3),we have 3-1 (x-0) y-3= 0-2 (0-) y-3-x y=-x+3 to -x3 andx varies from 0 to 2 2 That is, y varies from

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Step 3

Find the mass M of the regi...

M 8(x+y)A-8 (x+ y)
dydx
x/2
-x+3
2
* +3)
*(x/2)_ (x/2)]
x(-x+3)
dx
2
2
x2-6x9x2
-x2 +3x
2
2
-3x2 +9x
3
3x2
= 8
3
2
2
6
24
0
8[6]= 48
Thus, the mass of the region D is 48
help_outline

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M 8(x+y)A-8 (x+ y) dydx x/2 -x+3 2 * +3) *(x/2)_ (x/2)] x(-x+3) dx 2 2 x2-6x9x2 -x2 +3x 2 2 -3x2 +9x 3 3x2 = 8 3 2 2 6 24 0 8[6]= 48 Thus, the mass of the region D is 48

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