Cambridge Additional Mathematics

Integration (Chapter 15) 419

y = (x)f

t t+h

f(t) f(t + h)

x

enlarged strip

h

y

x

y = (x)f

a

t t+h

O b

Consider the narrow strip between x=t and x=t+h.

The area of this strip isA(t+h)¡A(t), but we also know it

must lie between a lower and upper rectangle on the interval

t 6 x 6 t+h of widthh.

area of smaller

rectangle

6 A(t+h)¡A(t) 6

area of larger

rectangle

Iff(x)is increasing on this interval then

hf(t) 6 A(t+h)¡A(t) 6 hf(t+h)

) f(t) 6

A(t+h)¡A(t)

h

6 f(t+h)

Equivalently, if f(x) is decreasing on this interval then f(t+h) 6

A(t+h)¡A(t)

h

6 f(t).

Taking the limit as h! 0 gives f(t) 6 A^0 (t) 6 f(t)

) A^0 (t)=f(t)

So, the area function A(t) must only differ from the

antiderivative of f(t) by a constant.

) A(t)=F(t)+c

Letting t=a, A(a)=F(a)+c

But from (1), A(a)=0

) c=¡F(a)

) A(t)=F(t)¡F(a)

Letting t=b,

Zb

a

f(x)dx=F(b)¡F(a)

This result is in fact true for all continuous functions f(x).

THE FUNDAMENTAL THEOREM OF CALCULUS

From the geometric argument above, the Fundamental Theorem of Calculus can be stated in two forms:

For a continuous function f(x), if we define the area function from x=a as A(t)=

Zt

a

f(x)dx,

then A^0 (x)=f(x).

or more commonly:

For a continuous function f(x) with antiderivative F(x),

Zb

a

f(x)dx=F(b)¡F(a).

PROPERTIES OF DEFINITE INTEGRALS

The following properties of definite integrals can all be deduced from the fundamental theorem of calculus:

²

Za

a

f(x)dx=0

²

Za

b

f(x)dx=¡

Zb

a

f(x)dx

²

Zb

a

cdx=c(b¡a) fcis a constantg

²

Zb

a

cf(x)dx=c

Zb

a

f(x)dx

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_15\419CamAdd_15.cdr Monday, 7 April 2014 3:58:21 PM BRIAN