Cambridge Additional Mathematics

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