Cambridge Additional Mathematics

(singke) #1
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

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