Page 398 - Schaum's Outline of Theory and Problems of Advanced Calculus

P. 398

CHAP. 15] GAMMA AND BETA FUNCTIONS 389

xdx y dy
ð ð 2
1 1
15.45. Prove that (a) 6 ¼ p ; ðbÞ 4 ¼ p ﬃﬃﬃ.
ﬃﬃﬃ
0 1 þ x 3 3 0 1 þ y 2 2
ð 2x
1 e 2
15.46. Prove that dx ¼ p ﬃﬃﬃ where a; b > 0.
1 ae 3x þ b 3 3 a 2=3 1=3
b
ð 2x
1 e 2
15.47. Prove that 3x dx ¼ p ﬃﬃﬃ
1 ðe þ 1Þ 9 3
[Hint: Diﬀerentiate with respect to b in Problem 15.46.]
15.48. Use the method of Problem 12.31, Chapter 12, to justify the procedure used in Problem 15.11.
DIRICHLET INTEGRALS
xy.
p ﬃﬃﬃﬃﬃﬃ
15.49. Find the mass of the region in the xy plane bounded by x þ y ¼ 1; x ¼ 0; y ¼ 0ifthe density is ¼
Ans: =24
x 2 y 2 z 2
15.50. Find the mass of the region bounded by the ellipsoid þ þ ¼ 1ifthe density varies as the square of
a 2 b 2 c 2
the distance from its center.
abck
2
2
2
Ans: ða þ b þ c Þ; k ¼ constant of proportionality
30
15.51. Find the volume of the region bounded by x 2=3 þ y 2=3 þ z 2=3 ¼ 1.
Ans: 4 =35
15.52. Find the centroid of the region in the ﬁrst octant bounded by x 2=3 þ y 2=3 þ z 2=3 ¼ 1.
Ans: x x ¼ y ¼ z ¼ 21=128
z
y
3
m
m
3
m
m
15.53. Show that the volume of the region bounded by x þ y þ z ¼ a , where m > 0, is given by 8f ð1=mÞg a .
2
3m ð3=mÞ
m
m
m
m
15.54. Show that the centroid of the region in the ﬁrst octant bounded by x þ y þ z ¼ a , where m > 0, is given
by
y z 3 ð2=mÞ ð3=mÞ a
x x ¼ y ¼ z ¼
4 ð1=mÞ ð4=mÞ
MISCELLANEOUS PROBLEMS
ð b
q
p
15.55. Prove that ðx aÞ ðb xÞ dx ¼ðb aÞ pþqþ1 Bð p þ 1; q þ 1Þ where p > 1; q > 1 and b > a.
a
[Hint: Let x a ¼ðb aÞy:]
ð 3 dx ð 7 p
ð7 xÞðx 3Þ dx.
15.56. Evaluate (a) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ðbÞ 4 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 ðx 1Þð3 xÞ 3
2
Ans: ðaÞ ; ðbÞ 2 f ð1=4Þg
3
ﬃﬃﬃ
p
2 p p
ﬃﬃﬃ ﬃﬃﬃ
3
2
15.57. Show that f ð1=3Þg ¼ p ﬃﬃﬃ .
3
ð1=6Þ
1 ð 1 x u 1 þ x v 1
uþv
15.58. Prove that Bðu; vÞ¼ dx where u; v > 0.
2 0 ð1 þ xÞ
[Hint: Let y ¼ x=ð1 þ xÞ:

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