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
√
(2x − 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 )
d y
(a) 0.677 (b) 0.674
(c) 0.675
10. A vehicle starts from rest and its velocity is
measured every second for 8 s, with values as
follows:
time t (s) velocity v (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 8.0 s is given by
8.0
0 v dt
Estimate this distance using Simpson’s rule,
giving the answer correct to 3 significant
figures.
[28.8 m]
11. A pin moves along a straight guide so that its
velocity v (m/s) when it is a distance x(m)
from the beginning of the guide at time t (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 determine the approximate total distance travelled
by the pin in the 4.0 s period.
[0.485 m]
(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
√
(2x − 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 )
d y
(a) 0.677 (b) 0.674
(c) 0.675
10. A vehicle starts from rest and its velocity is
measured every second for 8 s, with values as
follows:
time t (s) velocity v (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 8.0 s is given by
8.0
0 v dt
Estimate this distance using Simpson’s rule,
giving the answer correct to 3 significant
figures.
[28.8 m]
11. A pin moves along a straight guide so that its
velocity v (m/s) when it is a distance x(m)
from the beginning of the guide at time t (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 determine the approximate total distance travelled
by the pin in the 4.0 s period.
[0.485 m]