Cambridge Additional Mathematics

442 Applications of integration (Chapter 16)

Example 6 Self Tutor

Find the total area of the regions contained by y=f(x) and thex-axis for f(x)=x^3 +2x^2 ¡ 3 x.

f(x)=x^3 +2x^2 ¡ 3 x

=x(x^2 +2x¡3)

=x(x¡1)(x+3)

) y=f(x) cuts the x-axis at 0 , 1 , and¡ 3.

Total area

=

Z 0

¡ 3

(x^3 +2x^2 ¡ 3 x)dx¡

Z 1

0

(x^3 +2x^2 ¡ 3 x)dx

=

·

x^4

4

+

2 x^3

3

¡

3 x^2

2

̧ 0

¡ 3

¡

·

x^4

4

+

2 x^3

3

¡

3 x^2

2

̧ 1

0

=

¡

0 ¡¡ (^1114)
¢
¡
¡
¡ 127 ¡ 0
¢
=11^56 units^2

EXERCISE 16B

1 Find the exact value of the area bounded by:

a thex-axis and y=x^2 +x¡ 2

b thex-axis, y=e¡x¡ 1 , and x=2

c thex-axis and the part of y=3x^2 ¡ 8 x+4below thex-axis

d y= cosx, thex-axis, x=¼ 2 , and x=^32 ¼

e y=x^3 ¡ 4 x, thex-axis, x=1, and x=2

f y= sinx¡ 1 , thex-axis, x=0, and x=¼ 2

2 Find the area of the region enclosed by y=x^2 ¡ 2 x and y=3.

3 Consider the graphs of y=x¡ 3 and y=x^2 ¡ 3 x.

a Sketch the graphs on the same set of axes.

b Find the coordinates of the points where the graphs meet.

c Find the area of the region enclosed by the two graphs.

4 Determine the area of the region enclosed by y=

p

x and y=x^2.

5aOn the same set of axes, graph y=ex¡ 1 and y=2¡ 2 e¡x, showing axes intercepts and

asymptotes.

b Find algebraically the points of intersection of y=ex¡ 1 and y=2¡ 2 e¡x.

c Find the area of the region enclosed by the two curves.

6 Find the area of the region bounded by y=2ex, y=e^2 x, and x=0.

7 On the same set of axes, sketch y=2x and y=4x^2.

Find the area of the region enclosed by these functions.

-3 1 x

y

y = x + 2x - 3x 32

O

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_16\442CamAdd_16.cdr Monday, 7 April 2014 4:17:43 PM BRIAN