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Basic Mathematics-II
Notes
1
4. e 2 3x dx
0
f ( ) e 2 3x ,a 0,b 1 and nh b a 1 0 1
x
a
f ( ) f (0) e 2
( f a h f ( ) e 2 3h e 2 .e 3h
)
h
h
h
( f a 2 ) f (2 ) e 2 3h e 2 .e 3h
h
h
( f a 3 ) f (3 ) e 2 9h e 2 .e 9h
:
:
h
h
( f a n 1 ) f (1 n 1 ) e 2 3(n 1)h e 2 .e ( 3 n 1)h
b
h
f
Now, f ( )dx lim [ ( ) f (a h ) f (a 2 ) f (a 3 ) f (a n h
h
h
1 )]
x
a
h 0
a
1
f
e 2 3x dx lim [ (0) f ( ) f (2 ) f (3 ) ( f a n 1 )]
h
h
h
h
h
0 h 0
2
2
lim [e e 2 .e 3h e 2 .e 6h e 2 .e 9h e e 3(n 1)h ]
h
h 0
e 2 (e 3nh 1) h
lim h lim e 2 (e 3nh 1)
h 0 e 3nh 1 h 0 e 3h 1
3h 1
3
lim h e 2(e 1) [ nh 1]
h 0 e 3h 1 3
1 1 1 1
3
1 2
2
e 2 (e 1) (e e ) e
3 3 3 e
Example:
Evaluate the following definite integrals as limit of sums:
4 2
x
x
1. 2 x 2. 5 x
2 1
Solution:
4
1. 2 x
2
x
f ( ) 2 ,a 2,b 4 and nh b a 4 2 2
x
2
a
f ( ) f (2) e 4
h
2
( f a h f (2 h e 2 3h 2 .2 4.2 h
)
)
2
h
h
( f a 2 ) f (2 2 ) 2 2 2h 2 .2 2h 4.2 2h
h
h
( f a 3 ) f (2 3 ) 2 2 3h 22.2 3h 4.2 3h
:
:
2
h
( f a n 1 ) f (2 n 1 ) e 2 (n 1)h 2 .2 (n 1)h 4.2 (n 1)h
h
b
f
Now, f ( )dx lim [ ( ) f (a h ) f (a 2 ) f (a 3 ) f (a n 1 )]
h
h
x
h
a
h
h 0
a
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