274 Integrals
Problems 4.89
1 Find the pth moment about the line having the equation x = of the
lamina of constant areal density (mass per unit area) 5 which occupies the plane
region consisting of points (x,y) for whicha:x<=b, 0Sy<h. As.:p+1I(b-
2 A semicircular disk of radius a has its center at the origin and lies in the
right half-plane containing points (x,y) for which x >= 0. Find its centroid.11ns.:x=37r'y-0.
3 A homogeneous spherical ball of radius a has its center at the origin.
Find the centroid of the hemispherical part of the ball containing points for
which Ins.: z=$a,y =0,i=0.
4 Prove that the centroid of a right circular conical solid of height h has dis-
tance h/4 from the base of the solid.
5 Find the pth moment about the y axis of the region bounded by the x
axis, the line having the equation x = 1, and the graph of the equation y = xr,
it being supposed that r is a nonnegative constant. Ans.: 1/(p + r + 1).
6 Copy Figure 1.292 and let T be the triangular region bounded by the
triangle having vertices at A, B, C. Set up and evaluate all of the integrals
required for evaluation of ITI, the area of T, M,?0, the first moment of T
about the y axis, and MM,0, the first moment of T about the x axis. Then
use the formulas ITIx = M)0 and JT!y = My)a to find z and y. Remark:
The point (z,y) is the point (2h/3, 0). This shows that the centroid of T lies on
the median SID. More remarks can be made.
7 This problem involves hydrostatic forces which liquids exert upon surfaces
of bodies immersed in them. Before formulating our problem, we digress to eke
out some information. If an ordinary rectangular or cylindrical tank has hori-
zontal sections having area A square feet and if the tank is filled to depth d feet
with a liquid weighing w pounds per cubic foot, then the total weight of the con-
tents of the tank is wdA. If we divide this total weight wdA by the area -I of
the base of the tank, we obtain the number wd, which is the weight per square
foot that the base supports. This number wd, the product of w and the depth,
is called the pressure at depth d. This pressure wd is a scalar, the magnitude of
the force per unit area which the liquid exerts upon the flat horizontal base of
the tank. Our next task is to capture the idea that the jumble of words "pressure
in a gas and in a liquid is transmitted (sent across?) equally in all directions"
is often presumed to convey. To be very humble about this matter, we can
believe or perhaps even know that water will spurt from a hole in the bottom of a
tank of water and will spurt almost as vigorously from a hole near the bottom
of the tank but in a vertical side of the tank. Fortified by this idea, we can
cheerfully accept the ponderous physical principle or law which says that if a
plane region having area I is beneath the surface of a liquid, and if dl and d2
are numbers such that each point of the region has a depth d for which d, <= d 5
d2, then the force which the liquid exerts upon one side of this region is orthogonal
or normal or perpendicular to the region and there is a number d* such that