Cambridge Additional Mathematics

414 Integration (Chapter 15)

Historical note

#endboxedheading

The wordintegrationmeans “to put together into a whole”. Anintegralis the “whole” produced from

integration, since the areas f(xi)£w of the thin rectangular strips are put together into one whole

area.

The symbol

Z

is called anintegral sign. In the time ofNewtonandLeibnizit was the stretched out

letter s, but it is no longer part of the alphabet.

Example 1 Self Tutor

a Sketch the graph of y=x^4 for 06 x 61. Shade the area described by

Z 1

0

x^4 dx.

b Use technology to calculate the lower and upper rectangle sums fornequal subintervals where

n=5, 10 , 50 , 100 , and 500.

c Hence find

Z 1

0

x^4 dx to 2 significant figures.

abn AL AU

5 0 : 1133 0 : 3133

10 0 : 1533 0 : 2533

50 0 : 1901 0 : 2101

100 0 : 1950 0 : 2050

500 0 : 1990 0 : 2010

c When n= 500, AL¼AU¼ 0 : 20 ,to 2 significant figures.

) since AL<

Z 1

0

x^4 dx < AU,

Z 1

0

x^4 dx¼ 0 : 20

EXERCISE 15A.2

1aSketch the graph of y=

p

x for 06 x 61.

Shade the area described by

Z 1

0

p

xdx.

b Find the lower and upper rectangle sums for n=5, 10 , 50 , 100 , and 500.

c Hence find

Z 1

0

p

xdxto 2 significant figures.

2 Consider the region enclosed by y=

p

1+x^3 and thex-axis for 06 x 62.

a Write expressions for the lower and upper rectangle sums usingnsubintervals,

n 2 N.

b Find the lower and upper rectangle sums for n=50, 100 , and 500.

c Hence estimate

Z 2

0

p

1+x^3 dx.

GRAPHING

PACKAGE

AREA

FINDER

02. 04. 06. 08. 1

1

08.

06.

04.

02.

x

y

y=x¡¡¡^4

A=

Z 1

0

x^4 dx

O

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\414CamAdd_15.cdr Monday, 7 April 2014 3:57:46 PM BRIAN