434 Integration (Chapter 15)
Bernhard Riemann
x
y
ab
y = f(x)
A=
Zb
a
f(x)dx
O
AREA
FINDER
2
-2
2
4
y
6 x
semi-circle
O
Historical note
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Following the work of Newton and Leibniz, integration was rigorously formalised using limits by the
German mathematicianBernhard Riemann( 1826 - 1866 ).
Iff(x)> 0 on the intervala 6 x 6 b, we have seen that the area under the curve isA=
Zb
a
f(x)dx.
This is known as theRiemann integral.
Review set 15A
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1aSketch the region between the curve y=
4
1+x^2
and thex-axis for 06 x 61.
Divide the interval into 5 equal parts and display the 5 upper and lower rectangles.
b Use thearea findersoftware to find the lower and upper rectangle sums for
n=5, 50 , 100 , and 500.
c Give your best estimate for
Z 1
0
4
1+x^2
dx and compare this answer with¼.
2 The graph of y=f(x) is illustrated:
Evaluate the following using area interpretation:
a
Z 4
0
f(x)dx b
Z 6
4
f(x)dx
3 Integrate with respect tox:
a
4
p
x
b sin(4x¡5) c e^4 ¡^3 x
4 Find the exact value of:
a
Z¡ 1
¡ 5
p
1 ¡ 3 xdx b
Z ¼
2
0
cos
¡x
2
¢
dx
5 By differentiating y=
p
x^2 ¡ 4 , find
Z
x
p
x^2 ¡ 4
dx.
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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\434CamAdd_15.cdr Monday, 7 April 2014 4:00:10 PM BRIAN