Basic Engineering Mathematics, Fifth Edition

332 Basic Engineering Mathematics

40

30

20

10

5

0 1234

v (m/s)

t(s)

v  2 t^2  5

Figure 35.4

Problem 28. Sketch the graph

y=x^3 + 2 x^2 − 5 x−6 betweenx=−3and

x=2 and determine the area enclosed by

the curve and thex-axis

A table of values is produced and the graph sketched as

shown in Figure 35.5, in which the area enclosed by the

curve and thex-axisisshownshaded.

x − 3 − 2 − 1 0 1 2

y 0 4 0 − 6 − 8 0

y

x

6

23 22 21 01

y 5 x^3 1 2 x^2 2 5 x 2 6

2

Figure 35.5

Shaded area=

∫− 1

− 3

ydx−

∫ 2

− 1

ydx, the minus sign

before the second integral being necessary since the

enclosed area is below thex-axis. Hence,

shaded area=

∫− 1

− 3

(x^3 + 2 x^2 − 5 x− 6 )dx

−

∫ 2

− 1

(x^3 + 2 x^2 − 5 x− 6 )dx

=

[

x^4

4

+

2 x^3

3

−

5 x^2

2

− 6 x

]− 1

− 3

−

[

x^4

4

+

2 x^3

3

−

5 x^2

2

− 6 x

] 2

− 1

=

[{

1

4

−

2

3

−

5

2

+ 6

}

−

{

81

4

− 18 −

45

2

+ 18

}]

−

[{

4 +

16

3

− 10 − 12

}

−

{

1

4

−

2

3

−

5

2

+ 6

}]

=

[{

3

1

12

}

−

{

− 2

1

4

}]

−

[{

− 12

2

3

}

−

{

3

1

12

}]

=

[

5

1

3

]

−

[

− 15

3

4

]

= 21

1

12

or 21.08 square units

Problem 29. Determine the area enclosed by the

curvey= 3 x^2 +4, thex-axis and ordinatesx= 1

andx=4 by (a) the trapezoidal rule, (b) the

mid-ordinate rule, (c) Simpson’s rule and

(d) integration.

The curvey= 3 x^2 +4 is shown plotted in Figure 35.6.

The trapezoidal rule, the mid-ordinate rule and Simp-

son’s rule are discussed in Chapter 28, page 257.

(a) By the trapezoidal rule

area=

(

width of

interval

)[

1

2

(

first+last

ordinate

)

+

sum of

remaining

ordinates

]