Thomas Calculus 13th [Solutions]

Chapter 16 Additional and Advanced Exercises 1233
th
12. (a) Partition the sheet into small pieces. Let i be the area of the i piece and select a point ( xi , yi , z i ) in
th
th
the i piece. The mass of the i piece is approximately xi y i i . The work done by gravity in moving
th
the i piece to the xy -plane is approximately ( gxi yi i ) zi gxi yi zi i Work gxyz d .
S
(b)
2 2 1 1 x 2 2
gxyz d g xy(1 x y) 1 ( 1) ( 1) dA 3g xy x y xy dy dx
0 0
S
Rxy
3g 1 1 x
1 2 1 2 2 1 3 1
1 1 2 1 3 1 4
xy x y xy dx 3g x x x x
0 2 2 3 0
0 6 2 2 6
dx
3g 2 3 4 5
1
1 1 1 1 1 1 3g
x x x x 3g
12 6 6 30 0 12 30 20
(c) The center of mass of the sheet is the point ( x, y, z ) where z with M xy xyz d and
M
S
M xy d . The work done by gravity in moving the point mass at ( x, y, z ) to the
S
M xy
gMz gM gM
M xy gxyz d
S
13. (a) Partition the sphere
3g
20 .
M xy
xy-plane is
2 2 2
x y ( z 2) 1 into small pieces. Let i be the surface area of the
th
i piece
and let ( xi , yi , z i ) be a point on the i th
th
piece. The force due to pressure on the i piece is approximately
w(4 z i ) i . The total force on S is approximately w(4 z i ) i . This gives the actual force to be
i
w(4 z) d .
S
(b) The upward buoyant force is a result of the k -component of the force on the ball due to liquid pressure.
The force on the ball at ( x, y, z) is w(4 z)( n) w( z 4) n, where n is the outer unit normal at ( x, y, z).
Hence the k -component of this force is w( z 4) n k w( z 4) k n . The (magnitude of the) buoyant
force on the ball is obtained by adding up all these k -component s to obtain w( z 4) k n d .
(c) The Divergence Theorem says w( z 4) k n d div w( z 4) k dV w dV , where D is
S D D
2 2 2
x y ( z 2) 1 w( z 4) k n d w 1 dV 4 w,
the weight of the fluid if it were to
3
S
D
occupy the region D.
2 2
14. The surface S is z x y from z 1 to z 2. Partition S into small pieces and let i be the area of the
th
i piece. Let ( xi , yi , z i ) be a point on the i th
th
piece. Then the magnitude of the force on the i piece due to
liquid pressure is approximately F i w(2 z i ) i the total force on S is approximately
Fi w(2 z i ) i the actual force is
i
R
xy
S
2 2
x y
2 2 2 2
2 2
w(2 z) d w 2 x y 1
dA
x y x y
S
R
2 2 2 2
2 2 2 1 3 2 2 2w
4 2 w
2w 2 x y dA 2 w(2 r) r dr d 2w r r d d
0 1 0 3 1 0 3 3
xy
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