Answer to Question #196142 in Calculus for Angelo

Question #196142

Centroid of Plane Area


Locate the centroid of the region 𝑅 bounded by the given curves. Sketch the graph.


1.) 𝑅:𝑦=π‘₯^2, 𝑦=2π‘₯^2 βˆ’3π‘₯

2.) 𝑅:𝑦^2=4π‘₯, 𝑦=4, π‘¦βˆ’π‘Žπ‘₯𝑖𝑠

3.) 𝑅:π‘₯=2π‘¦βˆ’π‘¦^2, π‘¦βˆ’π‘Žπ‘₯𝑖𝑠

4.) 𝑅:𝑦=2π‘₯+1, π‘₯+𝑦=7, π‘₯=8


1
Expert's answer
2021-05-23T08:16:34-0400

1)"x=\\frac{\\iint\\limits_D xdxdy}{\\iint\\limits_D dxdy}=\\frac{\\int\\limits_0^3\\int\\limits_{2x^2-3x}^{x^2}xdydx}{\\int\\limits_0^3\\int\\limits_{2x^2-3x}^{x^2}dydx}=\\frac{\\int\\limits_0^3(x^2-2x^2+3x)xdx}{\\int\\limits_0^3(x^2-2x^2+3x)dx}=\\frac{6.75}{4.5}=1.5\\\\\ny=\\frac{\\iint\\limits_D ydxdy}{\\iint\\limits_D dxdy}=\\frac{0.5*\\int_0^3 ((x^2)^2-(2x^2-3x)^2)dx}{4.5}=\\frac{0.5*16.2}{4.5}=1.8"



2

"x=\\frac{\\iint\\limits_D xdxdy}{\\iint\\limits_D dxdy}=\\frac{\\int\\limits_0^44x\\sqrt{x}dx}{\\int\\limits_0^44\\sqrt{x}dx}=\\frac{256\/5}{64\/3}=2.4"

due to symmetry about the x-axis, y = 0


3

"x=\\frac{\\iint\\limits_D xdxdy}{\\iint\\limits_D dxdy}=\\frac{\\int_0^1 2*x*\\sqrt{1-x}dx}{\\int_0^1 2*\\sqrt{1-x}dx}=\\frac{8\/15}{4\/3}=0.4"

due to symmetry with respect to y = 1, y = 1


4

this is a triangle

"x=(1\/3)*(2+8+8)=6\\\\\ny=(1\/3)*(5 -1+17)=7"


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