. Differential and integral calculus. z-^) of the center of gravity of any given homogeneousbody, of volume v. If the body is symmetrical with reference to a plane, thisplane may be taken as the XY plane ; whence zx = o. If the body is symmetrical with reference to a straight line,this line may be taken as the X-axis; whence yx = o andzx = o. 243. To find the ce?iter of gravity of a circular arc. Let ABC be the arc and let OX be the axis of symmetry. 380 Integral Calculus Let (x,y) [= ODy DA\ be the coordinates of the extremity (A)of the arc. Since dv = ds = \ldx> + df we have, § 242, _/: x
. Differential and integral calculus. z-^) of the center of gravity of any given homogeneousbody, of volume v. If the body is symmetrical with reference to a plane, thisplane may be taken as the XY plane ; whence zx = o. If the body is symmetrical with reference to a straight line,this line may be taken as the X-axis; whence yx = o andzx = o. 243. To find the ce?iter of gravity of a circular arc. Let ABC be the arc and let OX be the axis of symmetry. 380 Integral Calculus Let (x,y) [= ODy DA\ be the coordinates of the extremity (A)of the arc. Since dv = ds = \ldx> + df we have, § 242, _/: x yfdx* + df- L-hffif* From the equation of the circle, x? + y2 — a2, we have T. dx y. — •>X X,= £A$) dy 2 yas Since 2y = chord AC, we see thatthe center of gravity of a circular arcis on its radius of symmetry and at adistance from its center equal to thefourth proportional between the arc, radius and chord. 244. 7b find the center of gravity of a circular dv = d2A = dxdy; hence, rx: 4 n xdxdy V*2 _ ** V««=P xdxdy 381 Mechanical Applications2 j V#2 — x?xdx Jx ~ A -f(^2-^A = %(a*-*y A ira2If the sector is a semicircle then A = —, and # = = o. 2 *, = 40 3^ 245. To find the center of gravity of the area bounded by aparabola, its axis and one of its ordinates.
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