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Unit 4: Definite Integral
h Notes
h
lim h 2 2cos 4 sin
0 h 2 8
sin
2
1
2cos sin sin2 sin
8 8 8 4 2
/2
2
3. sin xdx
/6
2
x
f ( )sin x , ,b and nh b a
6 2 2 6 3
1 cos
2 3
a
f ( ) f sin
6 2 2
1 cos 2h
2 3
)
( f a h f h sin h
6 6 2
2
h
( f a 2 ) f 2 sin 2h
6 6
1 cos 4h 1 cos 2.2h
3 3
2 2
2
h
( f a 3 ) f 3h sin 3h
6 6
1 cos 6h 1 cos 3.2h
3 3
2 2
2
h
( f a n 1 ) f n 1h sin n 1h
6 6
1 cos 2n 1h
3
2
b
h
)
f
a
h
h
x
h
f ( )dx lim | ( ) ( f a h ( f a 2 ) ( f a 3 ) .... ( f a n 1 )
Now, a h 0
/2
2
n
sin xdx lim h f f h f 2h f 3h f 1h
6
/6 6 6 6 6
1 cos 1 cos 2h 1 cos 2.2h 1 cos 3.2h 1 cos (n 1)2h
lim h 3 3 3 3 3
2 2 2 2 2
h
lim n cos cos 2h cos 2.2h cos 3.2h .. cos (n 1)2h
2 3 3 3 3 3
n 1 2h
h 0 cos 3 cos 2 2h sin n 2
h
lim n n
2 sin 2h
2
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