Calculate the centroid of region "R"?

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Answer 1 · 2018-07-13T13:58:12

#bar x = (int_0^2 x * y \ dx)/(int_0^2 y \ dx)#

# = (int_0^2 4 x - x^3 \ dx)/(int_0^2 4 - x^2 \ dx)#

# = ([ 2 x^2 - x^4/4 ]_0^2)/([ 4x - x^3/3 ]_0^2) = (4)/(16/3) = 3/4 = 0.75#

#bar y = (int_0^2 y/2 * y \ dx)/(int_0^2 y \ dx)#

# = (1/2int_0^2 16 - 8x^2 +x^4 \ dx)/(16/3)#

# = 1/2 ( [ 16x - 8/3x^3 +x^5/5 ]_0^2)/(16/3)#

#= 1/2* (32 - 64/3 + 32/5)/(16/3) = 8/5 = 1.6#