Cambridge Additional Mathematics

Integration (Chapter 15) 415

3 The integral

Z 3

¡ 3

e

¡x

2

2

dx is of considerable interest to statisticians.

a Use the graphing package to help sketch y=e

¡x

2

(^2) for ¡ 36 x 63.
b Calculate the upper and lower rectangular sums for the interval 06 x 63 using n= 2250.
c Use the symmetry of y=e
¡x
2
(^2) to find upper and lower rectangular sums for ¡ 36 x 60 for
n= 2250.
d Hence estimate
Z 3
¡ 3
e
¡x
2
(^2) dx.
How accurate is your estimate compared with
p
2 ¼?

Example 2 Self Tutor

Use graphical evidence and

known area facts to find:

a

Z 2

0

(2x+1)dx b

Z 1

0

p

1 ¡x^2 dx

a

Z 2

0

(2x+1)dx

=shaded area

=

¡1+5

2

¢

£ 2

=6

b If y=

p

1 ¡x^2 then y^2 =1¡x^2 and so x^2 +y^2 =1which is the equation of the unit

circle. y=

p

1 ¡x^2 is the upper half.

Z 1

0

p

1 ¡x^2 dx

=shaded area

=^14 (¼r^2 ) where r=1

=¼ 4

4 Use graphical evidence and known area facts to find:

a

Z 3

1

(1 + 4x)dx b

Z 2

¡ 1

(2¡x)dx c

Z 2

¡ 2

p

4 ¡x^2 dx

In many problems in calculus we know the rate of change of one variable with respect to another, but we

do not have a formula which relates the variables. In other words, we know

dy

dx

, but we need to knowy

in terms ofx.

B Antidifferentiation

AREA

FINDER

x

y

y=2x+1

2

5

3

1

(2 5),

O

-1 1

1

y

x

y = ~`1# -` #x 2

O

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_15\415CamAdd_15.cdr Monday, 7 April 2014 3:57:53 PM BRIAN