Thomas Calculus 13th [Solutions]

604 Chapter 8 Techniques of Integration
62. (a) The volume of the filled part equals the length
of the tank times the area of the shaded region
shown in the accompanying figure. Consider
a layer of gasoline of thickness dy located at
height y where r y r d . The width of
(b)
2 2
this layer is 2 r y . Therefore,
r d 2 2
A 2 r y dy and
r
r d 2 2
V L A 2L r y dy
r
2 2 2
r d 2 2 y r y 1 y
2L r y dy 2L
r sin
r
2 2 r
(We used FORMULA 29 with a r)
( d r) 2
2
1
2
2
2
1
2L 2rd d r sin d r r 2L d r 2rd d r sin d r
2 2 r 2 2 2 2 r 2
r d
r
2
63. The integrand f ( x)
x x is nonnegative, so the integral is maximized by integrating over the functions
entire domain, which runs from x 0 to x 1
1 1
2
1
1
1 2 1
2 2 2 2 1 2
2 1 x 1
x
x x dx x x dx 2 x x sin
0 0 2 2 2 2
1
2
0
(We used FORMULA 48 with a 1
2 )
1
x
1
2 2 1 1
x x sin (2x
1) 1 1
2 8 8 2 8 2 8
0
64. The integrand is maximized by integrating g( x) x 2x 2
x over the largest domain on which g is
nonnegative, namely [0, 2]
2
2 2
2 ( x 1)(2x 3) 2x x 1 1
x 2x x dx sin ( x 1)
0
6 2
0
(We used FORMULA 51 with a 1 )
1 1
2 2 2 2 2
CAS EXPLORATIONS
65. Example CAS commands:
Maple:
q1: Int( x*ln(x), x ); # (a)
q1 value( q1 );
q2 : Int( x^2*ln(x), x ); # (b)
q2 value( q2 );
q3 : Int( x^3*ln(x), x ); # (c)
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