Answer: mass (m) = 4 kg
center of mass coordinate: (15.75,4.5)
Step-by-step explanation: As a surface, a lamina has 2 dimensions (x,y) and a density function.
The region D is shown in the attachment.
From the image of the triangle, lamina is limited at x-axis: 0≤x≤2
At y-axis, it is limited by the lines formed between (0,0) and (2,1) and (2,1) and (0.3):
<u>Points (0,0) and (2,1):</u>
y = ![\frac{1-0}{2-0}(x-0)](https://tex.z-dn.net/?f=%5Cfrac%7B1-0%7D%7B2-0%7D%28x-0%29)
y = ![\frac{x}{2}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B2%7D)
<u>Points (2,1) and (0,3):</u>
y = ![\frac{3-1}{0-2}(x-0) + 3](https://tex.z-dn.net/?f=%5Cfrac%7B3-1%7D%7B0-2%7D%28x-0%29%20%2B%203)
y = -x + 3
Now, find total mass, which is given by the formula:
![m = \int\limits^a_b {\int\limits^a_b {\rho(x,y)} \, dA }](https://tex.z-dn.net/?f=m%20%3D%20%5Cint%5Climits%5Ea_b%20%7B%5Cint%5Climits%5Ea_b%20%7B%5Crho%28x%2Cy%29%7D%20%5C%2C%20dA%20%7D)
Calculating for the limits above:
![m = \int\limits^2_0 {\int\limits^a_\frac{x}{2} {2(x+y)} \, dy \, dx }](https://tex.z-dn.net/?f=m%20%3D%20%5Cint%5Climits%5E2_0%20%7B%5Cint%5Climits%5Ea_%5Cfrac%7Bx%7D%7B2%7D%20%20%7B2%28x%2By%29%7D%20%5C%2C%20dy%20%5C%2C%20dx%20%20%7D)
where a = -x+3
![m = 2.\int\limits^2_0 {\int\limits^a_\frac{x}{2} {(xy+\frac{y^{2}}{2} )} \, dx }](https://tex.z-dn.net/?f=m%20%3D%202.%5Cint%5Climits%5E2_0%20%7B%5Cint%5Climits%5Ea_%5Cfrac%7Bx%7D%7B2%7D%20%20%7B%28xy%2B%5Cfrac%7By%5E%7B2%7D%7D%7B2%7D%20%29%7D%20%5C%2C%20dx%20%20%7D)
![m = 2.\int\limits^2_0 {(-x^{2}-\frac{x^{2}}{2}+3x )} \, dx }](https://tex.z-dn.net/?f=m%20%3D%202.%5Cint%5Climits%5E2_0%20%7B%28-x%5E%7B2%7D-%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%7D%2B3x%20%29%7D%20%5C%2C%20dx%20%20%7D)
![m = 2.\int\limits^2_0 {(\frac{-3x^{2}}{2}+3x)} \, dx }](https://tex.z-dn.net/?f=m%20%3D%202.%5Cint%5Climits%5E2_0%20%7B%28%5Cfrac%7B-3x%5E%7B2%7D%7D%7B2%7D%2B3x%29%7D%20%5C%2C%20dx%20%20%7D)
![m = 2.(\frac{-3.2^{2}}{2}+3.2-0)](https://tex.z-dn.net/?f=m%20%3D%202.%28%5Cfrac%7B-3.2%5E%7B2%7D%7D%7B2%7D%2B3.2-0%29)
m = 2(-4+6)
m = 4
<u>Mass of the lamina that occupies region D is 4.</u>
<u />
Center of mass is the point of gravity of an object if it is in an uniform gravitational field. For the lamina, or any other 2 dimensional object, center of mass is calculated by:
![M_{x} = \int\limits^a_b {\int\limits^a_b {y.\rho(x,y)} \, dA }](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%20%5Cint%5Climits%5Ea_b%20%7B%5Cint%5Climits%5Ea_b%20%7By.%5Crho%28x%2Cy%29%7D%20%5C%2C%20dA%20%7D)
![M_{y} = \int\limits^a_b {\int\limits^a_b {x.\rho(x,y)} \, dA }](https://tex.z-dn.net/?f=M_%7By%7D%20%3D%20%5Cint%5Climits%5Ea_b%20%7B%5Cint%5Climits%5Ea_b%20%7Bx.%5Crho%28x%2Cy%29%7D%20%5C%2C%20dA%20%7D)
and
are moments of the lamina about x-axis and y-axis, respectively.
Calculating moments:
For moment about x-axis:
![M_{x} = \int\limits^a_b {\int\limits^a_b {y.\rho(x,y)} \, dA }](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%20%5Cint%5Climits%5Ea_b%20%7B%5Cint%5Climits%5Ea_b%20%7By.%5Crho%28x%2Cy%29%7D%20%5C%2C%20dA%20%7D)
![M_{x} = \int\limits^2_0 {\int\limits^a_\frac{x}{2} {2.y.(x+y)} \, dy\, dx }](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%20%5Cint%5Climits%5E2_0%20%7B%5Cint%5Climits%5Ea_%5Cfrac%7Bx%7D%7B2%7D%20%20%7B2.y.%28x%2By%29%7D%20%5C%2C%20dy%5C%2C%20dx%20%7D)
![M_{x} = 2\int\limits^2_0 {\int\limits^a_\frac{x}{2} {y.x+y^{2}} \, dy\, dx }](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%202%5Cint%5Climits%5E2_0%20%7B%5Cint%5Climits%5Ea_%5Cfrac%7Bx%7D%7B2%7D%20%20%7By.x%2By%5E%7B2%7D%7D%20%5C%2C%20dy%5C%2C%20dx%20%7D)
![M_{x} = 2\int\limits^2_0 { ({\frac{y^{2}x}{2}+\frac{y^{3}}{3})}\, dx }](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%202%5Cint%5Climits%5E2_0%20%7B%20%28%7B%5Cfrac%7By%5E%7B2%7Dx%7D%7B2%7D%2B%5Cfrac%7By%5E%7B3%7D%7D%7B3%7D%29%7D%5C%2C%20dx%20%7D)
![M_{x} = 2\int\limits^2_0 { ({\frac{x(-x+3)^{2}}{2}+\frac{(-x+3)^{3}}{3} -\frac{x^{3}}{8}-\frac{x^{3}}{24} )}\, dx }](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%202%5Cint%5Climits%5E2_0%20%7B%20%28%7B%5Cfrac%7Bx%28-x%2B3%29%5E%7B2%7D%7D%7B2%7D%2B%5Cfrac%7B%28-x%2B3%29%5E%7B3%7D%7D%7B3%7D%20-%5Cfrac%7Bx%5E%7B3%7D%7D%7B8%7D-%5Cfrac%7Bx%5E%7B3%7D%7D%7B24%7D%20%20%29%7D%5C%2C%20dx%20%7D)
![M_{x} = 2.(\frac{-9.x^{2}}{4}+9x)](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%202.%28%5Cfrac%7B-9.x%5E%7B2%7D%7D%7B4%7D%2B9x%29)
![M_{x} = 2.(\frac{-9.2^{2}}{4}+9.2)](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%202.%28%5Cfrac%7B-9.2%5E%7B2%7D%7D%7B4%7D%2B9.2%29)
![M_{x} = 18](https://tex.z-dn.net/?f=M_%7Bx%7D%20%3D%2018)
Now to find the x-coordinate:
x = ![\frac{M_{y}}{m}](https://tex.z-dn.net/?f=%5Cfrac%7BM_%7By%7D%7D%7Bm%7D)
x = ![\frac{63}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B63%7D%7B4%7D)
x = 15.75
For moment about the y-axis:
![M_{y} = \int\limits^2_0 {\int\limits^a_\frac{x}{2} {2x.(x+y))} \, dy\,dx }](https://tex.z-dn.net/?f=M_%7By%7D%20%3D%20%5Cint%5Climits%5E2_0%20%7B%5Cint%5Climits%5Ea_%5Cfrac%7Bx%7D%7B2%7D%20%20%7B2x.%28x%2By%29%29%7D%20%5C%2C%20dy%5C%2Cdx%20%7D)
![M_{y} = 2.\int\limits^2_0 {\int\limits^a_\frac{x}{2} {x^{2}+yx} \, dy\,dx }](https://tex.z-dn.net/?f=M_%7By%7D%20%3D%202.%5Cint%5Climits%5E2_0%20%7B%5Cint%5Climits%5Ea_%5Cfrac%7Bx%7D%7B2%7D%20%20%7Bx%5E%7B2%7D%2Byx%7D%20%5C%2C%20dy%5C%2Cdx%20%7D)
![M_{y} = 2.\int\limits^2_0 {y.x^{2}+x.{\frac{y^{2}}{2} } } \,dx }](https://tex.z-dn.net/?f=M_%7By%7D%20%3D%202.%5Cint%5Climits%5E2_0%20%7By.x%5E%7B2%7D%2Bx.%7B%5Cfrac%7By%5E%7B2%7D%7D%7B2%7D%20%7D%20%7D%20%5C%2Cdx%20%7D)
![M_{y} = 2.\int\limits^2_0 {x^{2}.(-x+3)+\frac{x.(-x+3)^{2}}{2} - {\frac{x^{3}}{2}-\frac{x^{3}}{8} } } \,dx }](https://tex.z-dn.net/?f=M_%7By%7D%20%3D%202.%5Cint%5Climits%5E2_0%20%7Bx%5E%7B2%7D.%28-x%2B3%29%2B%5Cfrac%7Bx.%28-x%2B3%29%5E%7B2%7D%7D%7B2%7D%20-%20%7B%5Cfrac%7Bx%5E%7B3%7D%7D%7B2%7D-%5Cfrac%7Bx%5E%7B3%7D%7D%7B8%7D%20%20%7D%20%7D%20%5C%2Cdx%20%7D)
![M_{y} = 2.\int\limits^2_0 {\frac{-9x^3}{8}+\frac{9x}{2} } \,dx }](https://tex.z-dn.net/?f=M_%7By%7D%20%3D%202.%5Cint%5Climits%5E2_0%20%7B%5Cfrac%7B-9x%5E3%7D%7B8%7D%2B%5Cfrac%7B9x%7D%7B2%7D%20%20%20%7D%20%5C%2Cdx%20%7D)
![M_{y} = 2.({\frac{-9x^4}{32}+9x^{2})](https://tex.z-dn.net/?f=M_%7By%7D%20%3D%202.%28%7B%5Cfrac%7B-9x%5E4%7D%7B32%7D%2B9x%5E%7B2%7D%29)
![M_{y} = 2.({\frac{-9.2^4}{32}+9.2^{2}-0)](https://tex.z-dn.net/?f=M_%7By%7D%20%3D%202.%28%7B%5Cfrac%7B-9.2%5E4%7D%7B32%7D%2B9.2%5E%7B2%7D-0%29)
63
To find y-coordinate:
y = ![\frac{M_{x}}{m}](https://tex.z-dn.net/?f=%5Cfrac%7BM_%7Bx%7D%7D%7Bm%7D)
y = ![\frac{18}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B18%7D%7B4%7D)
y = 4.5
<u>Center mass coordinates for the lamina are (15.75,4.5)</u>