Cambridge Additional Mathematics

Applications of integration (Chapter 16) 443

8 Sketch the circle with equation x^2 +y^2 =9.

a Explain why the upper half of the circle has equation y=

p

9 ¡x^2.

b Hence, determine

Z 3

0

p

9 ¡x^2 dx without actually integrating the function.

9 Find the area enclosed by the function y=f(x) and thex-axis for:

a f(x)=x^3 ¡ 9 x b f(x)=¡x(x¡2)(x¡4) c f(x)=x^4 ¡ 5 x^2 +4.

10 Answer theOpening Problemon page

11 a Explain why the total area shaded isnot

equal to

Z 7

1

f(x)dx.

b Write an expression for the total shaded

area in terms of integrals.

12 The illustrated curves are y= cos(2x) and

y=^12 +^12 cos(2x).

a Identify each curve as C 1 or C 2.

b Determine the coordinates of A, B, C, D, and E.

c Show that the area of the shaded region is¼ 2 units^2.

13 Explain why the area between two functions f(x) and g(x) on the interval a 6 x 6 b is given by

A=

Zb

a

jf(x)¡g(x)jdx.

14 The shaded area is 1 unit^2.

Findb, correct to 4 decimal places.

15 The shaded area is 6 aunits^2.

Find the exact value ofa.

y

O 13 57 x

y = f(x)

y

-a O a x

y=x +2 2

438.

y

x

AE

B C D

C 1

C 2

O

y

O b x

y = ~`x

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_16\443CamAdd_16.cdr Monday, 7 April 2014 4:20:25 PM BRIAN