Higher Engineering Mathematics

SOME APPLICATIONS OF INTEGRATION 375

H

By firstly determining the points of intersection the

range ofx-values has been found. Tables of values

are produced as shown below.

x − 3 − 2 − 1012

y=x^2 + 1 10 5 2125

x − 302

y= 7 −x 1075

A sketch of the two curves is shown in Fig. 38.3.

Shaded area=

∫ 2

− 3

(7−x)dx−

∫ 2

− 3

(x^2 +1)dx

=

∫ 2

− 3

[(7−x)−(x^2 +1)]dx

=

∫ 2

− 3

(6−x−x^2 )dx

=

[

6 x−

x^2

2

−

x^3

3

] 2

− 3

=

(

12 − 2 −

8

3

)

−

(

− 18 −

9

2

+ 9

)

=

(

7

1

3

)

−

(

− 13

1

2

)

= 20

5

6

square units

Figure 38.3

Problem 3. Determine by integration the area

bounded by the three straight linesy= 4 −x,

y= 3 xand 3y=x.

Each of the straight lines are shown sketched in

Fig. 38.4.

Shaded area

=

∫ 1

0

(

3 x−

x

3

)

dx+

∫ 3

1

[

(4−x)−

x

3

]

dx

=

[

3 x^2

2

−

x^2

6

] 1

0

+

[

4 x−

x^2

2

−

x^2

6

] 3

1

=

[(

3

2

−

1

6

)

−(0)

]

+

[(

12 −

9

2

−

9

6

)

−

(

4 −

1

2

−

1

6

)]

=

(

1

1

3

)

+

(

6 − 3

1

3

)

=4 square units

Figure 38.4

Now try the following exercise.

Exercise 148 Further problems on areas

under and between curves

1. Find the area enclosed by the curve
   y=4 cos 3x, thex-axis and ordinatesx= 0
   andx=

π

6

[1^13 square units]

2. Sketch the curvesy=x^2 +3 andy= 7 − 3 x
   and determine the area enclosed by them.
   [20^56 square units]


3. Determine the area enclosed by the three
   straight linesy= 3 x,2y=xandy+ 2 x=5.
   [2^12 square units]