440 Applications of integration (Chapter 16)
Discovery
Zb
a
f(x)dx and areas
Does
Zb
a
f(x)dx always give us an area?
What to do:
1 Find
Z 1
0
x^3 dx and
Z 1
¡ 1
x^3 dx.
2 Using a graph, explain why the first integral in 1 gives an area, whereas the second integral does not.
3 Find
Z 0
¡ 1
x^3 dx and explain why the answer is negative.
4 Show that
Z 0
¡ 1
x^3 dx+
Z 1
0
x^3 dx=
Z 1
¡ 1
x^3 dx.
5 Find
Z¡ 1
0
x^3 dx and interpret its meaning.
6 Suppose f(x) is a function such that f(x) 60 for all a 6 x 6 b. Suggest an expression for
the area between the curve and the function for a 6 x 6 b.
If two functions f(x) and g(x) intersect at
x=a and x=b, and f(x)>g(x) for all
a 6 x 6 b, then the area of the shaded region
between their points of intersection is given by
A=
Zb
a
[f(x)¡g(x)]dx.
Alternatively, if the upper and lower functions
are y=yU and y=yL respectively, then
the area is
A=
Zb
a
[yU¡yL]dx.
We can see immediately that if f(x) is the
x-axis f(x)=0, then the enclosed area
B The area between two functions
a b x
y
O
y = g(x)
y = f(x) = 0
O A
y = g(x)ory = yL
y = f(x)ory = yU
is
Zb
a
[¡g(x)]dx or ¡
Zb
a
g(x)dx.
cyan magenta yellow black
(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_16\440CamAdd_16.cdr Monday, 7 April 2014 4:19:51 PM BRIAN