442 Higher Engineering Mathematics
(c) the mid-ordinate rule, (d) Simpson’s rule. Give
answers correct to 3 decimal places.
6.
∫ 4
1
4
x^3
dx (Use 6 intervals)
[
(a) 1. 875 (b) 2. 107
(c) 1. 765 (d) 1. 916
]
7.
∫ 6
2
1
√
( 2 x− 1 )
dx (Use 8 intervals)
[
(a) 1. 585 (b) 1. 588
(c) 1. 583 (d) 1. 585
]
In Problems 8 and 9 evaluate the definite integrals
using (a) the trapezoidal rule, (b) the mid-ordinate
rule, (c) Simpson’s rule. Use 6 intervals in each
case and give answers correct to 3 decimal places.
8.
∫ 3
0
√
( 1 +x^4 )dx
[
(a) 10. 194 (b) 10. 007
(c) 10. 070
]
9.
∫ 0. 7
0. 1
1
√
( 1 −y^2 )
dy
[
(a) 0. 677 (b) 0. 674
(c) 0. 675
]
- A vehicle starts from rest and its velocity is
measured every second for 8s, with values as
follows:
timet(s) velocityv(ms−^1 )
0 0
1.0 0.4
2.0 1.0
3.0 1.7
4.0 2.9
5.0 4.1
6.0 6.2
7.0 8.0
8.0 9.4
The distance travelled in 8.0s is given by∫
8. 0
0 vdt
Estimate this distance using Simpson’s rule,
giving the answer correct to 3 significant
figures. [28.8m]
- A pin moves along a straight guide so that its
velocityv(m/s) when it is a distancex(m)
from the beginning of the guide at timet(s) is
given in the table below.
t(s) v(m/s)
0 0
0.5 0.052
1.0 0.082
1.5 0.125
2.0 0.162
2.5 0.175
3.0 0.186
3.5 0.160
4.0 0
Use Simpson’s rule with 8 intervals to deter-
mine the approximate total distance travelled
by the pin in the 4.0s period. [0.485 m]