Page 235 - Schaum's Outline of Theory and Problems of Advanced Calculus

226 MULTIPLE INTEGRALS [CHAP. 9

x
ð ð t ð z
FðuÞ du dt; ðx uÞFðuÞ du: Then
Let IðxÞ¼ JðxÞ¼
0 0 0
z z
ð ð
FðuÞ du; FðuÞ du
0 0
I ðxÞ¼ J ðxÞ¼
0 0
using Leibnitz’s rule, Page 186. Thus, I ðxÞ¼ J ðxÞ, and so IðxÞ JðxÞ¼ c, where c is a constant. Since
0
0
Ið0Þ¼ Jð0Þ¼ 0, c ¼ 0, and so IðxÞ¼ JðxÞ.
The result is sometimes written in the form
x x x
ð ð ð
2
FðxÞ dx ¼ ðx uÞFðuÞ du
0 0 0
The result can be generalized to give (see Problem 9.58)
x
ð ð x ð x ð x
n 1 n 1
FðxÞ dx ¼ ðx uÞ FðuÞ du
0 0 0 ðn 1Þ! 0
Supplementary Problems

DOUBLE INTEGRALS
2
9.23. (a) Sketch the region r in the xy plane bounded by y ¼ 2x and y ¼ x.(b)Find the area of r.(c)Find
the polar moment of inertia of r assuming constant density .
Ans.(b) 2 ; ðcÞ 48 =35 ¼ 72M=35, where M is the mass of r.
3
4
9.24. Find the centroid of the region in the preceding problem. Ans. x x ¼ ; y ¼ 1
y
5
p ffiffiffiffiffiffi
3 4 y
ð ð
9.25. Given ðx þ yÞ dx dy. (a) Sketch the region and give a possible physical interpretation of the
y¼0 x¼1
double integral. (b)Interchange the order of integration. (c) Evaluate the double integral.
ð 2 ð 4 x 2
Ans: ðbÞ ðx þ yÞ dy dx; ðcÞ 241=60
x¼1 y¼0
ð 2 ð x x ð 4 ð 2 x
9:26: Show that sin dy dx þ sin dy dx ¼ 4ð þ 2Þ :
ffiffi 2y p ffiffi 2y 3
x¼1 y¼ x x¼2 y¼ x
p
9.27. Find the volume of the tetrahedron bounded by x=a þ y=b þ z=c ¼ 1 and the coordinate planes.
Ans. abc=6
3
2
9.28. Find the volume of the region bounded by z ¼ x þ y ; z ¼ 0; x ¼ a; x ¼ a; y ¼ a; y ¼ a.
4
Ans.8a =3
9.29. 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 .
2
Ans.(a) 112 6 14 Ma , where M ¼ mass; x y z 7 a 2
45 a ¼ 15 (b) x ¼ y ¼ 0; z ¼ 15
TRANSFORMATION OF DOUBLE INTEGRALS
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðð q
2
2
2
2
2
9.30. Evaluate x þ y dx dy, where r is the region x þ y @ a . Ans. 2 3 a 3
r