Additional Mathematics
Chapter 10: Calculus II - Integration
Chapter Assessment
1
2
(i)
Given that
(ii)
Find
dy
 4 x3  6 x 2  5 , find y.
dx
[3]
  2 x  3 dx .
[3]
A curve passes through the point (1, 1) and has gradient function
dy
 4x  5 .
dx
Find the equation of the curve.
3
[4]
Find the value of the following.
3
(i)
  4 x  1 dx
[3]
0
1
(ii)
  3x
2
 x  1 dx
[4]
2
4
Find the area of the shaded region between the curve y  x 2  x  7 , the line x = 3 and
the axes.
[4]
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Additional Mathematics
5
The solid side of a small rocking toy has the shape given in Fig. 5.
The curve ABC has equation y  x 2  4 x  4 , D has coordinates (6, 16), the side DE has
equation 2y = 3x + 14 and the side EA is the y-axis.
D
E
C
A
B
Fig.5
Find the area of the shaded region representing the side of the rocking toy.
Units are centimetres.
6
(i)
Calculate the coordinates of the points A and B where the curve y = 6x  x2 meets the
straight line y = x + 4.
[4]
(ii)
Sketch the curve and the line and shade the area that is completely enclosed by the
curve and the line.
(iii) Calculate this shaded area.
7
[7]
[3]
[5]
A curve has equation y = x3(4  x).
(i)
Find the values of x for which y = 0.
[2]
(ii)
Find the coordinates of the turning points.
[5]
(iii) Calculate the area enclosed by the curve and the x-axis.
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Additional Mathematics
8
2 roads, AB and CD, meet at a junction. AB runs due East-West and CD runs North-South.
At the junction there is an arm from A that turns northwards to meet the other road at C and
an arm from A that turns southwards to meet the other road at D. The shape enclosed is a
grassed area.
The plan of this junction may be modelled as follows:
Road CD is the y-axis and road AB is parallel to the x-axis.
D is the origin.
The vertices of the grassed area are A, C and D.
1
The equation of the upper arm from A to D is y   x 2  20 x  200  .
5
1
The equation of the lower arm from A to D is y  x  20  x  .
5
A is at the point which is a maximum for one curve and a minimum for the other curve.
C
A
B
D
(i)
Find the coordinates of the points A and C.
[5]
(ii)
Find the area of the grass.
[5]
Total: 60
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