Thomas Calculus 13th [Solutions]

Chapter 11 Additional and Advanced Exercises 875
2 2 2
Also, M y xdm x dA a t sin t a 1 cost dt
0
3 2
2 2
a t 2t cos t t cos t sin t 2sin t cos t sin tcos
t dt
o
2 3 2
3 1 2 1
2 cos
2 3
a t 2cost 2t sin t t cos 2t t sin 2t cos t sin t t 3 a .
2 4 8 4 3
0
M
2 3
Thus
y 3 a
x a a,
5 a is the center of mass.
M 2
6
3 a
tan 2 tan 1
25. 2 1 tan tan 2 1 ;
1 tan tan
the curves will be orthogonal when tan is
undefined, or when tan
2
r f ( ) g ( )
1
2 1
1 r 1
2 tan g ( )
r
f ( )
26.
r
4
sin 4
sin
dr 3
4
d
sin cos tan tan
3
sin cos
4 4
4 4 4 4
27. r 2 a sin 3 dr 6 cos3 tan r 2asin 3 1 tan 3 ;
d
a 6a
cos3 3
when , tan 1 tan
6 3 2 2
dr
d
28. (a) (b)
r
1 2
1 r
dr
d
1
tan | 1
lim tan
2
from the right as the spiral winds in
2
around the origin.
29.
3 cos
tan 1
cot is 1 at ; tan sin
3 sin
3 3 2 tan is 3 at
cos
slopes is 1, the tangents are perpendicular
; since the product of these
3
30.
tan
r
dr
d
a 1 cos
a sin
is 1 at
2 4
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