Calculus: Analytic Geometry and Calculus, with Vectors

684 Iterated and multiple integrals

line x = xa. Ans.:

M(p)

xa

dx

yi+m2(x-xl)

x-xo ixi yi+m.a(x-xt)

X3 y,+m,(x-x,)

+ fSa dx yx+'ma(x-x2) (xx - x dy.

14 Develop computational skill by simplifying the answer to Problem 13

for the case in which S(x,y) = 1 and p = 0 so the answer is numerically equal to

the area of the lamina. Ans.:

[(x,y2 - x2Yi) - (xIy2 - xaYi) + (xzya - x3Y2)] or.

Yi 1

Y2 1

Y3 1

15 Figure 13.492 shows two parallel rods of lengths a and b. The rod on

the left has linear density (mass per unit length) 81(t) at distance t from its lower

(x - xo)PS(x,y) dy

xl

x2

x3

Y

b

a

pt

------------

u

01 _D X

Figure 13.492

end, and the rod on the right has linear density 62(u) at distance u from its lower

end. We undertake to learn about the total or resultant gravitational force F on

the left-hand rod that is produced by the right-hand rod. Assume that particles

of mass m I and m2 at points P, and P2 attract each other with a force of magnitude

Gm1m2/IP1P212, where G is a gravitational constant, and the actual force pulling

the particle at P, toward P2 is obtained by multiplying this magnitude by P,P2/

lP,P21, the unit vector which has its tail at Pl and points toward P2. Derive

the formula

51(t) At 62(U) Au

G

[D2 + (u - t)2] [Di + (u - t)J]

for the force which an element (or subset) of the rod on the right exerts upon an

element of the rod on the left. Then derive the formula

G51(t) At b a2(u)[Di + Cu- t)9]du

fo [D2 + (u - t)2]%

for the force which the whole rod on the right exerts upon the element of the rod

on the left. Then derive the formula

F = G f a Si(t) 'it fb 12(u)[Di + (U- t)j]du.

0 o [D2 + (u - t)2]%

16 Two identical slender rods having constant linear density S occupy the

intervals -a < x 5 -e and e <x S a of an x axis. It is supposed that 0 < e