Determine the ycoordinate of the centroid of the shaded area

Determine the y-coordinate of the centroid of the shaded area. tolerance is +/-1 in the 3rd significant digit

Solution

Slope of line = (b - 0.62b) / (b - 0) = 0.38

Equation of line is y = 0.38*x

x = y^2 /b

y = sqrt(bx)

Taking a vertical strip at distance x of width dx parallel to y-axis we get

Area of strip dA = [sqrt(bx) - 0.38x] dx

Area of shaded area A = Integral dA

= Integral [sqrt(bx) - 0.38x] dx.........x varies from x = 0 to x = b

= sqrt(b)*(x^1.5) / 1.5 - 0.19*x^2...........x varies from x = 0 to x = b

= sqrt(b)*(b^1.5) / 1.5 - 0.19*b^2

= 143/300 b^2

Yc = Integral y dA / Area

Integral y dA = Integral [(sqrt(bx))^2 - (0.38x)^2]/2 dx...........x varies from x = 0 to x = b

= Integral [bx/2 - 0.0722*x^2] dx...........x varies from x = 0 to x = b

= [(b/4)*x^2 - (0.0722 / 3)*x^3]...........x varies from x = 0 to x = b

= (b/4)*b^2 - (0.0722 / 3)*b^3

= (2.7112 / 12) *b^3

Yc = (2.7112 / 12) *b^3 / (143 / 300 *b^2)

= 0.474 b