Calculus: Analytic Geometry and Calculus, with Vectors

13.1 Iterated integrals 657

The integral in (1) will exist as a Riemann-Cauchy integral and will have the
value Y if

(6) V=Tlim 237(x+2+

lim

(r2+r4+r6

r8 F ..1

rz_ \1z 2z 32 42

With the aid of the basic fact that E(1/n2) = 7r2/6, prove that Y = a2/6. Hint:

Supposing that 0 < r < 1, let the last series in (6) converge to g(r) and begin

by showing that g(r) < 7r2/6. Then compare g(r) with 1/k2 - e/2

k=1

14 Prove that the formula

r

f b dx f df(x,Y) dy=f ab dx f d g(x,Y) dy + f b dx f d h(x,Y)dy

is valid provided (i) f(x,y) = g(x,y) + h(x,y) when a 5 x < b and c < y < d

and (ii) the integrals in the right member exist. Hint: Let

F(x) = J

d

f(x,Y) dy, G(x) = f d g(x,y) dy, H(x) = fed h(x,Y) dY

and use known facts about simple integrals.

15 Let

(1) F(x,y,z) = P(x,y,z)i + Q(x,y,z)j + R(x,y,z)k,

where the functions are continuous over some spherical ball B with center at

(xo,yo,zo), be the force on a particle when the particle is at the point (x,y,z).

Show that the work Wj(x,y,z) done by the force in moving the particle along line

segments from (xo,yo,zo) to (xo,y,zo) and then to (xo,y,z) and finally to the point

(x,y,z) in the ball is

(2) Wi(x,y,z) = (

V

loQ(xo,a,zo) dQ + LO R(xo,y,7) dY + f P(a,y,z)z da

(3)

awl- - = P(x,y,z)

ax

Show that the work W2(x,y,z) done by the force in moving the particle along

line segments from (xo,yo,zo) to (x,yo,zo) and then to (x,yo,z) and then to (x,y,z) is

(4) W2(x,y,z) = f xz P(a,yo,zo) da +fZaR(x,yo,7) d7 + fao do

and that

as

(5) (^22) = Q(x,y,z)
Show that the work W$(x,y,z) done by the force in moving the particle along line