Thomas Calculus 13th [Solutions]

Section 6.4 Areas of Surfaces of Revolution 467
2
2 2 3 1 6 1 1 6
ds dx dy y 1 dy y 1 dy y 1 1 dy
4 y 2 16 y 2 16 y
23.
3 6 6
3 1
2
3 1
4 y
4 y
2 2 3 1
2 4 1 2
S 2 y ds 2 y y dy 2 y y dy
1 1 4 y
1 4
y dy y dy ;
3 3
3
5 2
y 1
2 1 y 2 32 1 1 1 2 31 1 2 (8 31 5) 253
5 4 5 8 5 4 5 8 40 20
1
24.
y 2
2 /2
2
cos x dy
dy
sin sin 2 (cos ) 1 sin
dx
x dx
x S /2
x x dx
25.
2 2 dy 2 2
1/2
y a x 1 a x ( 2 x)
2
dy
2
dx 2 2 dx 2 2
a x
a
x
a 2
a a a
S 2 2 2 2 2
2 1 x 2 2 2
a a x dx 2 2
a x
a a x x dx a
a dx a x
2
2 a[ a ( a)] (2 a)(2 a) 4 a
x
x
2
a
26.
y 2 2 2
2 2 2 2
r dy r dy r 2 h r 1 r 2
h r h r 2 r h r
h
h x dx h dx S 2
0 h x dx 2
0 h x dx 2 h 2
h h h h 0
x dx
2 h
2
2 r 2 2 2 2 2 2 2
h r x r h r h r h r
2 2 2
h
0 h
2
2
27. The area of the surface of one work is S d
2 1 dx .
c
x dy
dy Now, 2 2 2 2 2
x y 16 x 16 y
2 2
dx y dx y
7 2
2 2 y
7 2 2 2
; S 2 16 y 1 dy 2 16 y y dy
dy 2 2
2 2
16 y
dy 2 2
16 y
16 16 y 16
7 2
2 16 dy 32 9 288 904.78 cm . The enamel needed to cover one surface of one wok is
16
3 3
V S 0.5 mm S 0.05 cm (904.78)(0.05) cm 45.24 cm . For 5000 woks, we need
3
5000 V 5000 45.24 cm (5)(45.24) L 226.2L 226.2 liters of each color are needed.
28.
2
2 2 dy
y r x
1 2x x dx x
r x r x
dx 2 2 2 2 2 dy 2 2
r x
a h 2 2 2
a h
2 r x x dx 2 r dx 2 rh , which is independent of a.
a
a
2
;
a h 2 2
2
S 2 r x 1 x dx
a
2 2
r x
29.
2
2 2 dy
y R x
1 2x x dx x
R x R x
dx 2 2 2 2 2 dy 2 2
R x
a h 2 2 2
a h
2 R x x dx 2 R dx 2 Rh
a
a
2
;
a h 2 2
2
S 2 R x 1 x dx
a
2 2
R x
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