Thomas Calculus 13th [Solutions]

354 Chapter 5 Integration
45.
3
f ( x) 2x
Let x 1 0 1 and c i
i i x . The right-hand sum is
n n n
n
n
3
3 n
2
2c 1 2 i 1 2 3 2 n( n 1)
i i
n n n 4 4 2
i 1 i 1
n i 1 n
2 1
2 2 2
1
n
2 n ( n 2n 1) n 2n
1
n
n
2
3
. Thus, lim 2c
1
4 2
4n
2n
2
i
n
n
i 1
1
2 1
n
n
2
lim 1 .
n
2 2
46.
2 3
f ( x)
x x
2 2
2 5n 5 4n 6n 2 n 2n
1
2n 2 2
3n 4n
lim 2
n
5
5 4 6 2 2 1
2 1
n
n
n
n
n
2
2 3 4
5 4
6 2
1
2 1
n n n
2 n
n
2
0 ( 1)
Let x 1 and c 1 1 i
i i x .
n n n
n
2 3
The right-hand sum is c 1
i ci n
i 1
n
2 3 n
1 i 1 i 1
2 3
2 5i 4i i 1
n n n
n 2 3 n
i 1
i 1 n n
n
2 3 n n n n
2 5i 4i i 2 5 4 2 1 3
n 2 3 4
2 3 4
i 1 n n n n i i i
i 1 n i 1 n i 1 n i 1
2 5 n( n 1)
2
( n)
4 n( n 1)(2n 1) 1 n( n 1)
n 2
n 2 3 6 4
n
n 2
5 n
2 3
2 . Thus, lim c 1
2 3 4
i ci n
n
i 1
2 5 4 1 7 .
2 3 4 12
5.3 THE DEFINITE INTEGRAL
1.
2 2
0 x dx 2. 0 3
1 2x dx 3. 5 2
7 ( x 3 x)
dx
4.
4 1
1 x dx 5. 3 1
2 1 x dx 6. 1 2
4 x
0
dx
7.
0
/4 (sec x)
dx 8. /4
(tan x)
dx
0
9. (a)
(c)
(d)
(e)
(f )
2
1 5
g( x) dx 0
(b) g( x) dx g( x) dx 8
2
5 1
2 2
3 f ( x) dx 3 f ( x) dx 3( 4) 12
1 1
5 5 2
f ( x) dx f ( x) dx f ( x) dx 6 ( 4) 10
2 1 1
5
1 [ f ( x) g( x)]
dx 5
5
f ( x)
dx g( x) dx 6 8 2
1
1
5
[4 ( ) ( )]
1 f x g x dx 5 5
4 f ( x) dx g( x)
dx 4(6) 8 16
1 1
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