Cambridge Additional Mathematics

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.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_16\440CamAdd_16.cdr Monday, 7 April 2014 4:19:51 PM BRIAN