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MULTIPLE INTEGRALS
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(b) Letting , the volume of the sphere thus obtained is 2a3 4 1 cos  a3 3 3
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9.20. a (b)
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Find the centroid of the region in Problem 9.19. Use the result in (a) to nd the centroid of a hemisphere.
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" " " " " (a) The centroid x; y; z is, due to symmetry, given by x y 0 and z dV Total moment about xy plane " z Total mass  dV Since z r cos  and  is constant the numerator is =2 a =2 4 a r   sin  cos  d d 4 r cos  r2 sin  dr d d 4   0  0 r 0  0  0 4 r 0 =2 a4 sin  cos  d d a4 =2
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0  0 2
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 4 2 sin    d a sin  2  0 4  0
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The denominator, obtained by multiplying the result of Problem 9.19(a) by , is 2 a3 1 cos . 3 Then " z 2
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3 4 2 1 4 a sin 3 1 cos a
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3 a 1 cos : 8
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" 8 (b) Letting =2; z 3 a.
MISCELLANEOUS PROBLEMS ' ' 1 & 1 1 & 1 x y 1 x y 1 dy dx , (b) dx dy . 9.21. Prove that (a) 2 2 x y 3 x y 3 0 0 0 0
a 1 & 1
' ' 1 & 1 x y 2x x y dy dx dy dx 3 3 x y 0 x y 0 0  ' 1 & 1  2x 1 dy dx 3 2 x y 0 0 x y 1 1  x 1  dx 2 x y y 0 0 x y  1 dx 1 1 1  2 x 1 0 2 0 x 1
' 1 & 1 y x 1 dx dy and (b) This follows at once on formally interchanging x and y in (a) to obtain 3 2 0 0 x y then multiplying both sides by 1. This example shows that interchange in order of integration may not always produce equal results. A su cient condition under which the order may be interchanged is that the double integral over the x y corresponding region exists. In this case dx dy, where r is the region x y 3 0 @ x @ 1; 0 @ y @ 1 fails to exist because of the discontinuity of the integrand at the origin. integral is actually an improper double integral (see 12).
x & t 9.22. Prove that
' x F u du dt x u F u du.
x & t
MULTIPLE INTEGRALS ' F u du dt; z
[CHAP. 9
Let I x
J x z
x u F u du:
Then z F u du
I 0 x
F u du;
J 0 x
using Leibnitz s rule, Page 186. Thus, I 0 x J 0 x , and so I x J x c, where c is a constant. Since I 0 J 0 0, c 0, and so I x J x . The result is sometimes written in the form x x x F x dx2 x u F u du
0 0 0
The result can be generalized to give (see Problem 9.58) x x x x 1 F x dxn x u n 1 F u du n 1 ! 0 0 0 0
Supplementary Problems
DOUBLE INTEGRALS 9.23. (a) Sketch the region r in the xy plane bounded by y2 2x and y x. (b) Find the area of r. (c) Find the polar moment of inertia of r assuming constant density . Ans. (b) 2 ; c 48=35 72M=35, where M is the mass of r. 3 Find the centroid of the region in the preceding problem. p
3 4 y
9.24. 9.25.
Ans.
" 5 " x 4;y 1
Given
y 0 x 1
x y dx dy.
(a) Sketch the region and give a possible physical interpretation of the (c) Evaluate the double integral.
double integral. Ans: b 2
(b) Interchange the order of integration. 4 x2 x y dy dx; c 241=60 4
x 1 y 0
2 9:26: Show that
x sin p
x 1 y x
x dy dx 2y
p y x
x 4  2 : dy dx 2y 3
Find the volume of the tetrahedron bounded by x=a y=b z=c 1 and the coordinate planes. Ans. abc=6 Find the volume of the region bounded by z x3 y2 ; z 0; x a; x a; y a; y a. Ans. 8a4 =3 Find (a) the moment of inertia about the z-axis and (b) the centroid of the region in Problem 9.28 assuming a constant density . 7 " " " Ans. (a) 112 a6  14 Ma2 , where M mass; (b) x y 0; z 15 a2 45 15
TRANSFORMATION OF DOUBLE INTEGRALS q x2 y2 dx dy, where r is the region x2 y2 @ a2 . 9.30. Evaluate