436 Higher Engineering Mathematics
Problem 1. (a) Use integration to evaluate,
correct to 3 decimal places,
∫ 3
1
2
√
x
dx(b) Use the
trapezoidal rule with 4 intervals to evaluate the
integral in part (a), correct to 3 decimal places.
(a)
∫ 3
1
2
√
x
dx=
∫ 3
1
2 x−
1
(^2) dx
⎡
⎢
⎣
2 x
(− 1
2
)
- 1
−
1
2
1
⎤
⎥
⎦
3
1
[
4 x
1
2
] 3
1
= 4
[√
x
] 3
1 =^4
[√
3 −
√
1
]
=2.928,correct to 3 decimal places
(b) The range of integration is the difference between
the upper and lower limits, i.e. 3− 1 =2. Using
the trapezoidal rule with 4 intervals gives an inter-
val widthd=
3 − 1
4
= 0 .5 and ordinates situated
at 1.0, 1.5, 2.0, 2.5 and 3.0. Corresponding values
of
2
√
x
are shown in the table below, each correct
to 4 decimal places (which is one more decimal
place than required in the problem).
x
2
√
x
1.0 2.0000
1.5 1.6330
2.0 1.4142
2.5 1.2649
3.0 1.1547
From equation (1):
∫ 3
1
2
√
x
dx≈( 0. 5 )
{
1
2
( 2. 0000 + 1. 1547 )
- 6330 + 1. 4142 + 1. 2649
}
=2.945,correct to 3 decimal places
This problemdemonstrates that even withjust 4 inter-
vals a close approximation to the true value of 2.928
(correct to 3 decimal places) is obtained using the
trapezoidal rule.
Problem 2. Use the trapezoidal rule with 8
intervals to evaluate,
∫ 3
1
2
√
x
dxcorrect to 3
decimal places.
With 8 intervals, the width of each is
3 − 1
8
i.e. 0.25
giving ordinates at 1.00, 1.25, 1.50, 1.75, 2.00, 2.25,
2.50, 2.75 and 3.00. Corresponding values of
2
√
x
are
showninthetablebelow.
x
2
√
x
1.00 2.0000
1.25 1.7889
1.50 1.6330
1.75 1.5119
2.00 1.4142
2.25 1.3333
2.50 1.2649
2.75 1.2060
3.00 1.1547
From equation (1):
∫ 3
1
2
√
x
dx≈( 0. 25 )
{
1
2
( 2. 000 + 1. 1547 )+ 1. 7889
- 6330 + 1. 4142 + 1. 2649
- 6330 + 1. 5119 + 1. 4142
- 3333 + 1. 2649 + 1. 2060
}
=2.932,correct to 3 decimal places.
This problem demonstrates that the greater the number
of intervals chosen (i.e. the smaller the interval width)
the more accurate will be the value of the definite inte-
gral. The exact value is found when the number of
intervalsis infinite, whichis, ofcourse, what the process
of integration is based upon.
Problem 3. Use the trapezoidal rule to evaluate
∫ π
2
0
1
1 +sinx
dxusing 6 intervals. Give the answer
correct to 4 significant figures.
- 3333 + 1. 2649 + 1. 2060