Page 123 - Schaum's Outline of Theory and Problems of Advanced Calculus

114 INTEGRALS [CHAP. 5

ð 2 dx ð 1
2
5.63. Evaluate to 3 decimal places using numerical integration: (a) 2 ; ðbÞ cosh x dx.
1 1 þ x 0
Ans. (a)0.322, (b)1.105.
APPLICATIONS
5.64. Find the (a)area and (b) moment of inertia about the y-axis of the region in the xy plane bounded by
y ¼ sin x,0 @ x @ and the x-axis, assuming unit density.
2
Ans. (a)2, (b) 4
2
5.65. Find the moment of inertia about the x-axis of the region bounded by y ¼ x and y ¼ x,ifthe density is
proportional to the distance from the x-axis.
Ans. 1 8 M, where M ¼ mass of the region.
3
5.66. (a) Show that the arc length of the catenary y ¼ cosh x from x ¼ 0to x ¼ ln 2 is .(b) Show that the length
4
ffiffiffi
p
of arc of y ¼ x 3=2 ,2 @ x @ 5is 343 2 2 11 3=2 .
27
5.67. Show that the length of one arc of the cycloid x ¼ að sin Þ, y ¼ að1 cos Þ, ð0 @ @ 2 Þ is 8a.
2
2
2
2
5.68. Prove that the area bounded by the ellipse x =a þ y =b ¼ 1is ab.
5.69. (a)(Disk Method) Find the volume of the region obtained by revolving the curve y ¼ sin x,0 @ x @ ,
2
about the x-axis. Ans: ðaÞ =2
(b)(Disk Method) Show that the volume of the frustrum of a paraboloid obtained by revolving
ð b
p ffiffiffiffiffiffi k 2 2
kx,0 < a @ x @ b, about the x-axis is ðb a Þ. (c)Determine the volume
f ðxÞ¼ kx dx ¼ 2
a p ffiffiffi p ffiffiffi
2
obtained by rotating the region bound by f ðxÞ¼ 3, gðxÞ¼ 5 x on 2 @ x @ 2.(d)(Shell Method)
Aspherical bead of radius a has a circular cylindrical hole of radius b, b < a,through the center. Find the
volume of the remaining solid by the shell method. (e)(Shell Method) Find the volume of a solid whose
2
2
2
outer boundary is a torus (i.e., the solid is generated by orbiting a circle ðx aÞ þ y ¼ b about the y-axis
(a > b).
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
p
2
5.70. Prove that the centroid of the region bounded by y ¼ a x , a @ x @ a and the x-axis is located at
ð0; 4a=3 Þ.
5.71. (a)If ¼ f ð Þ is the equation of a curve in polar coordinates, show that the area bounded by this curve and
1 ð 2
2
the lines ¼ 1 and ¼ 2 is d . (b)Find the area bounded by one loop of the lemniscate
2
2
¼ a cos 2 . 2 1
Ans. (b) a 2
ð q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
5.72. (a)Prove that the arc length of the curve in Problem 5.71(a)is þðd =d Þ d .(b)Find the length
of arc of the cardioid ¼ að1 cos Þ. 1
Ans. (b)8a
MISCELLANEOUS PROBLEMS
5.73. Establish the mean value theorem for derivatives from the first mean value theorem for integrals. [Hint: Let
f ðxÞ¼ F ðxÞ in (4), Page 93.]
0
ð 4 dx ð 3 dx ð 1 dx
5.74. Prove that (a) lim p ffiffiffiffiffiffiffiffiffiffiffi ¼ 4; ðbÞ lim p ffiffiffi ¼ 6; ðcÞ lim p ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ and give a geo-
!0þ 0 4 x !0þ 3 x !0þ 0 1 x 2 2
metric interpretation of the results.
4 dx 3 dx 1 dx
ð ð ð
[These limits, denoted usually by p ffiffiffiffiffiffiffiffiffiffiffi ; p ffiffiffi and p ffiffiffiffiffiffiffiffiffiffiffiffiffi respectively, are called impro-
0 4 x 0 3 x 0 1 x 2
per integrals of the second kind (see Problem 5.29) since the integrands are not bounded in the range of
integration. For further discussion of improper integrals, see Chapter 12.]
M dx
ð ð 2
5 x
5.75. Prove that (a) lim x e dx ¼ 4! ¼ 24; ðbÞ lim p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ .
M!1 0 !0þ 1 xð2 xÞ 2