Cambridge Additional Mathematics

Applications of integration (Chapter 16) 451

5 OABC is a rectangle and the two shaded

regions are equal in area. Findk.

6 The shaded region has area^12 unit^2.

Find the value ofm.

7 Find the area of the region enclosed byy=x and y= sin

³

¼x

2

́

.

8 A boat travelling in a straight line has its engine turned off at time t=0. Its velocity at time

tseconds thereafter is given by v(t)=

100

(t+2)^2

ms¡^1.

a Find the initial velocity of the boat, and its velocity after 3 seconds.

b Discuss v(t) as t!1.

c Sketch the graph of v(t) againstt.

d Find how long it takes for the boat to travel 30 metres from when the engine is turned off.

e Find the acceleration of the boat at any timet.

f Show that

dv

dt

=¡kv

3

(^2) , and find the value of the constantk.
m
y
O ¼ x
y=sinx
y
x
A B
C
2
k
O
y=x +k 2
4037 Cambridge
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Y:\HAESE\CAM4037\CamAdd_16\451CamAdd_16.cdr Monday, 7 April 2014 4:18:48 PM BRIAN