Take the radius of synunetry as the axis of x and the arc in the

plane, XY, as in Fig. 18. Then dl : dy::r : x, or dF'=cfl=— , and

X

hence

~ I ~ I I ^^^

That is, the center of gravity is on the radius of symmetry at a
distance from the center equal to a fourth proportional to the arc,
radius, and chord.
\£f(x)dx]
Ex. AS. SolveEx. 47byi=i=^2^-| ^.
Ex.49. Solve Ex. 47 by usmg dl=^V{d3fi+dy^) and 3^+y^=r^.
Ex. Find the center of gravity of a symmetrical arc of the
cycloid.
Take the cycloid as in Fig. 19, and use the equation P=8ra:.
Answer f x=ix'.
Ex. 51. Find the center of gravity of a triangle by integra-
tion.
Take a vertex at the origin and the opposite side parallel to F,



'%.
PROBLEMS



173



as in Fig. 20, and let y'sax and y"^bx be the equations of the
two other sides. Then we may write dS==(y"-'y')dXf and

__[f(6-a)^
[Jo (6-«)«ic

[\ (6-a)xdx] 2

This result agrees with that obtained in Ex. 41, the medians
intersecting at two-thirds of the distance from any vertex to the
middle of the opposite side.
FiQ. 18.
Fig. 19.
Ex. 52. Find the center of gravity of a circular sector.
