06 03 integration

7/25/2019 06 03 Integration

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1. The area of the enclosed region shown in the diagram is defined by

yx2+ 2,yax+ 2, where a> 0.

x

y

a0

2

This region is rotated 360 about thex-ais to form a solid of re!olution. "ind, in terms of a, the

!olume of this solid of re!olution.

Working:

Answers#

$$$$$$$$$$$$$$$$..

(Total 4 marks)

1


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2. %sing the substitution u& 2'

x+ ', or otherwise, find the integral

+'2

' xx

dx.

Working:

Answers#

$$$$$$$$$$$$$$$$..

(Total 4 marks)

3. (hen air is released from an inflated balloon it is found that the rate of decrease of the !olume

of the balloon is )ro)ortional to the !olume of the balloon. This can be re)resented by the

differential e*uation tvdd

& kv, where vis the !olume,tis the time and kis the constant of

)ro)ortionality.

a f the initial !olume of the balloon is v0, find an e)ression, in terms of k, for the !olume

of the balloon at time t.

b "ind an e)ression, in terms of k, for the time when the !olume is 20v

.

Working:

Answers#

a

$$$$$$$$$$$$$$$

$..

b

$$$$$$$$$$$$$$.....

2


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.....

(Total 4 marks)

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4. /onsider the functionf #x xx2for ' xk, where ' k3.

a 1etch the gra)h of the functionf.

(3)

b "ind the total finite area enclosed by the gra)h off, thex-ais and the linex& k.

(4)

(Total 7 marks)

5. The area between the gra)h ofy & ex

and thex-ais fromx& 0 tox& k k> 0 is rotated through

360 about thex-ais. "ind, in terms of kand e, the !olume of the solid generated.

Working:

Answers#

....

$$$$$$$$$$$$$$..........

(Total 4 marks)

6. "ind the real number k> ' for which

+

k

x'

2''

dx& 23

.

Working:

Answers#

....

$$$$$$$$$$$$$$..........

4


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(Total 4 marks)

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7. n the diagram,PTQis an arc of the )arabolay& a2x

2, where ais a )ositi!e constant, and

PQRSis a rectangle. The area of the rectanglePQRSis e*ual to the area between the arcPTQof

the )arabola and thex-ais.

y

x

S R

QP

O

y = a x2 2

T

"ind, in terms of a, the dimensions of the rectangle.

Working:

Answers#

....

$$$$$$$$$$$$$$.....

.....

(Total 4 marks)

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8. /onsider the functionfkx &

=>0,0

0,n'

x

xkxxx

, where

a "ind the deri!ati!e offkx,x> 0.

(2)

b "ind the inter!al o!er whichfx is increasing.

The gra)h of the functionfkx is shown below.

(2)

y

x0 3

c i 1how that the stationary )oint offkx is atx& ek'.

ii 4nex-interce)t is at 0, 0."ind the coordinates of the otherx-interce)t.

(4)

d "ind the area enclosed by the cur!e and thex-ais.

(5)

e "ind the e*uation of the tangent to the cur!e at .

(2)

f 1how that the area of the triangular region created by the tangent and the

coordinate aes is twice the area enclosed by the cur!e and thex-ais.

(2)

g 1how that thex-interce)ts offkx for consecuti!e !alues of kform a geometric se*uence.

(3)

(Total 20 marks)

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9. "ind the !alues of a > 0, such that +

2

2'

'a

a x dx& 0.22.

Working:

Answers#

....

$$$$$$$$$$$$$$.....

.....

(Total 3 marks)

10. 5etfx= ln x7 3x

2, 0.7 x < 2, x a, x 8 a, are !alues ofx for whichfxis not

defined.

a i 1etch the gra)h offx, indicating on your setch the number of 9eros of fx!

1how also the )osition of any asym)totes.

(2)

ii "ind all the 9eros offx, that is, sol!efx & 0.

(3)

b "ind the exact !alues of a and !

(3)

c "indfx, and indicate clearly wherefxis not defined.

(3)

d "ind the exact !alue of thex-coordinate of the local maimum offx, for 0 x < '.7.

:ou may assume that there is no )oint of infleion.

(3)

e Write down the definite integral that re)resents the area of the region enclosedbyfx

and thex-ais. ;o not e!aluate the integral.

(2)

(Total 16 marks)

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11. /alculate the area bounded by the gra)h ofy = x sin x2and thex-ais, betweenx = 0 and the

smallest )ositi!ex-interce)t.

Working:

Answers#

....

$$$$$$$$$$$$$$.....

.....

(Total 3 marks)

12. 1ol!e the differential e*uationxy x

y

d

d

& ' +y2,gi!en thaty = 0 whenx = 2.

Working:

Answers#

....

$$$$$$$$$$$$$$.....

.....

(Total 3 marks)

13. uniform rod of length "metres is )laced with its ends on two su))orts , < at the same

hori9ontal le!el.

xm e t r e s

" m e t r e s

y x, -

3

06 03 integration

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