Cambridge Additional Mathematics

Integration (Chapter 15) 421

EXERCISE 15C

1 Use the fundamental theorem of calculus to find the area between:

a thex-axis and y=x^3 from x=0to x=1

b thex-axis and y=x^2 from x=1to x=2

c thex-axis and y=

p

x from x=0to x=1.

2 Use the fundamental theorem of calculus to show that:

a

Za

a

f(x)dx=0and explain the result graphically

b

Zb

a

cdx=c(b¡a) wherecis a constant

c

Za

b

f(x)dx=¡

Zb

a

f(x)dx

d

Zb

a

cf(x)dx=c

Zb

a

f(x)dx wherecis a constant

e

Zb

a

[f(x)+g(x)]dx=

Zb

a

f(x)dx+

Zb

a

g(x)dx

3 Use the fundamental theorem of calculus to find the area between thex-axis and:

a y=x^3 from x=1to x=2

b y=x^2 +3x+2from x=1to x=3

c y=

p

x from x=1to x=2

d y=ex from x=0to x=1: 5

e y=

1

p

x

from x=1to x=4

4aUse the fundamental theorem of calculus to show that

Zb

a

(¡f(x))dx=¡

Zb

a

f(x)dx

b Hence show that if f(x) 60 for allxon

a 6 x 6 b then the shaded area=¡

Zb

a

f(x)dx.

c Calculate the following integrals, and give graphical interpretations of your answers:

i

Z 1

0

(¡x^2 )dx ii

Z 1

0

(x^2 ¡x)dx iii

Z 0

¡ 2

3 xdx

d Use graphical evidence and known area facts to find

Z 2

0

³

¡

p

4 ¡x^2

́

dx.

y

ab x

y = f(x)

O

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_15\421CamAdd_15.cdr Monday, 7 April 2014 3:58:35 PM BRIAN