1.2. Sets 11(c)Ck={(x, y):0≤x^2 +y^2 ≤ 1 /k},k=1, 2 , 3 ,....1.2.8.For every one-dimensional setC, define the functionQ(C)=∑
Cf(x),
wheref(x)=(^23 )(^13 )x,x=0, 1 , 2 ,..., zero elsewhere. IfC 1 ={x:x=0, 1 , 2 , 3 }
andC 2 ={x:x=0, 1 , 2 ,...}, findQ(C 1 )andQ(C 2 ).
Hint: Recall thatSn=a+ar+···+arn−^1 =a(1−rn)/(1−r) and, hence, it
follows that limn→∞Sn=a/(1−r) provided that|r|<1.
1.2.9.∫ For every one-dimensional setCfor which the integral exists, letQ(C)=
Cf(x)dx,wheref(x)=6x(1−x), 0<x<1, zero elsewhere; otherwise, letQ(C)
be undefined. IfC 1 ={x:^14 <x<^34 },C 2 ={^12 },andC 3 ={x:0<x< 10 }, find
Q(C 1 ),Q(C 2 ), andQ(C 3 ).
1.2.10.For every two-dimensional setCcontained inR^2 for which the integral
exists, letQ(C)=
∫∫
C(x(^2) +y (^2) )dxdy.IfC 1 ={(x, y):− 1 ≤x≤ 1 ,− 1 ≤y≤ 1 },
C 2 ={(x, y):− 1 ≤x=y≤ 1 },andC 3 ={(x, y):x^2 +y^2 ≤ 1 }, findQ(C 1 ),Q(C 2 ),
andQ(C 3 ).
1.2.11.LetCdenote the set of points that are interior to, or on the boundary of, a
square with opposite vertices at the points (0,0) and (1,1). LetQ(C)=
∫∫
Cdy dx.
(a)IfC⊂Cis the set{(x, y):0<x<y< 1 }, computeQ(C).
(b)IfC⊂Cis the set{(x, y):0<x=y< 1 }, computeQ(C).
(c)IfC⊂Cis the set{(x, y):0<x/ 2 ≤y≤ 3 x/ 2 < 1 }, computeQ(C).
1.2.12.LetCbe the set of points interior to or on the boundary of a cube with
edge of length 1. Moreover, say that the cube is in the first octant with one vertex
at the point (0∫∫∫ , 0 ,0) and an opposite vertex at the point (1, 1 ,1). LetQ(C)=
Cdxdydz.
(a)IfC⊂Cis the set{(x, y, z):0<x<y<z< 1 }, computeQ(C).
(b) IfCis the subset{(x, y, z):0<x=y=z< 1 }, computeQ(C).
1.2.13.LetCdenote the set{(x, y, z):x^2 +y^2 +z^2 ≤ 1 }. Using spherical coordi-
nates, evaluate
Q(C)=
∫∫∫
C
√
x^2 +y^2 +z^2 dxdydz.
1.2.14.To join a certain club, a person must be either a statistician or a math-
ematician or both. Of the 25 members in this club, 19 are statisticians and 16
are mathematicians. How many persons in the club are both a statistician and a
mathematician?
1.2.15.After a hard-fought football game, it was reported that, of the 11 starting
players, 8 hurt a hip, 6 hurt an arm, 5 hurt a knee, 3 hurt both a hip and an arm,
2 hurt both a hip and a knee, 1 hurt both an arm and a knee, and no one hurt all
three. Comment on the accuracy of the report.