This is what people have long known as the "white tongue line" (in the math manual).

This problem is a simple generalized integral problem.

Establish a rectangular coordinate system with OA as the y axis and OE as the x axis.

The rectangular coordinate equation of the tongue line is y = 8a 2/(x 2+4a 2),

The parametric equation of the tongue line is: x = 2atanθ, y = 2a (cos θ) 2.

When the radius of the circle is a = 1, it is the red curve Y = 8/(x 2+4) in this question.

This is just a generalized "closed curve".

When the density is constant (1), due to symmetry, the abscissa of the center of mass is 0;

The vertical coordinate formula of the center of mass of the curved trapezoid is (1/2) ∫.

"Closed Graph" area surrounded by curve and x axis:

s =∫& lt； -∞，+∞& gt； 8dx/(4+x^2)= 16∫& lt； 0，+∞& gt； dx/(4+x^2)

= 8[arctan(x/2)]& lt； 0，+∞& gt； = 4π,

I =( 1/2)∫& lt； -∞，+∞& gt； [8/(4+x^2)]^2dx =∫& lt； 0，+∞& gt； 64dx/(4+x^2)^2

Let x = 2tanu, then dx = 2 (secu) 2du.

I =∫& lt； 0，π/2 & gt； 8(cosu)^2du = 4∫& lt； 0，π/2 & gt； ( 1+cos2u)du

= 2[2u+sin2u]& lt； 0，π/2 & gt； = 2π,

The ordinate of the center of mass is I/S = 1/2.

The centroid of a "closed graph" surrounded by a curve and the X axis is (0, 1/2).