MULTIPLE INTEGRALS AND THEIR APPLICATIONS
421
44.80
Find the center of mass (x, y) of the plate cut from the parabola y
2 = Sx by its latus rectum x = 2 if the
density is numerically equal to the distance from the latus rectum.
See Fig. 44-41.
By symmetry,
The mass
The moment about the y-axis is given by
Hence,
Thus, the center of mass is
Fig. 44-41
Fig. 44-42
44.81
Find the center of mass of a plate in the form of the upper half of the cardioid r = 2(1 + cos 0) if the density is
numerically equal to the distance from the pole.
See Fig. 44-42. The mass
The moment about the x-axis is
Hence,
The moment about the y-axis is
So the center of mass is
44.82
Find the center of mass of the first-quadrant part of the disk of radius a with center at the origin, if the density
function is y.
The mass
The
moment about the A:-axis is
Hence,
The moment about the y-axis is
Therefore,
Hence, the center of mass is
Hence,