Higher Engineering Mathematics

440 INTEGRAL CALCULUS

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 inte-

grals 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

]

10. A vehicle starts from rest and its velocity is
    measured every second for 8 s, with values as
    follows:

timet(s) velocityv(ms−^1 )

00

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 vdt

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 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)

00

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 4.0 s period. [0.485 m]