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