Radius of gyration; centroid by integration y = kx²

Radius of gyration

The radius of gyration of an area about an axis is the distance k from the axis at which the whole area, if concentrated at a point, would give the same moment of inertia: I = A k², k = √(I/A). It measures how far the area is spread from the axis, and it is used in column design (slenderness ratio).

Centroid by direct integration

Reading of the figure: the area lies between the y axis, the top line y = b and the curve y = kx² (which goes from O to the point (a, b)). It is the area above the curve. So k = b/a². Use a vertical strip at x with height from y = kx² to y = b, i.e. h = b − kx².

Area A = ∫₀ᵃ (b − kx²) dx = ab − k a³/3 = ab − ab/3 = 2ab/3

x̄ ∫ x dA = ∫₀ᵃ x (b − kx²) dx = b a²/2 − k a⁴/4 = a²b/2 − a²b/4 = a²b/4 x̄ = (a²b/4) / (2ab/3) = 3a/8

ȳ (strip centroid at the middle of the strip, (b + kx²)/2) ∫ ȳ dA = ∫₀ᵃ ½ (b + kx²)(b − kx²) dx = ½ ∫₀ᵃ (b² − k² x⁴) dx = ½ (ab² − ab²/5) = 2ab²/5 ȳ = (2ab²/5) / (2ab/3) = 3b/5

Answer: centroid at (3a/8, 3b/5) from O. (Check: the area under the curve alone has centroid (3a/4, 3b/10); taking moments of the two complementary areas about the y axis: (ab/3)(3a/4) + (2ab/3)(3a/8) = a²b/4 + a²b/4 = a²b/2 = (ab)(a/2) ✓.)