Cambridge Additional Mathematics

(singke) #1
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

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