The y-coordinate of the centroid is given by the ratio of two
definite integrals;
yc = ∫ydm/∫dm, where dm = a density function δ evaluated
over dA.
For the uniform plate, δ does not change with position in the plate.
yc = ∫yδdA/∫δdA = ∫ydA/∫dA.
dA is a horizontal slice of the plate with dimensions xdy.
Solving the parabola for x,
y = 1.050x2
x = ± √(y/1.050), where the negative value corresponds to the left half of the parabola and the positive to the right half.
dA = (√(y/1.050) - - √(y/1.050))dy = 2(√(y/1.050))dy
The limits of integration are from zero to 1.480, the top of the plate.
The integral in the numerator has an extra y.
Using a TI-83 Plus,
∫ydA = ∫2y√(y/1.050)dy = 2.080 m3
∫dA = ∫2√(y/1.050)dy = 2.343 m2
∫ydA/∫dA = 2.080/2.343 = .888 m up from rounded tip of plate.
A uniform plate of height 1.480 m is cut in the form of a parabolic section....
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