A LEVEL P1

QUESTION'S

INTEGRATION
 QUESTIONS 1 TOPIC:8 INTEGRATION

1
√
1 Evaluate
(3x + 1) dx. [4]
0

18
The diagram shows part of the graph of y = and the normal to the curve at P (6, 3). This normal
x
meets the x-axis at R. The point Q on the x-axis and the point S on the curve are such that PQ and SR
are parallel to the y-axis.

(i) Find the equation of the normal at P and show that R is the point (4 12 , 0). [5]

(ii) Show that the volume of the solid obtained when the shaded region PQRS is rotated through
360◦ about the x-axis is 18π . [4]

dy 6
3 A curve is such that =√ and P (3, 3) is a point on the curve.
dx (4x − 3)

(i) Find the equation of the normal to the curve at P, giving your answer in the form ax + by = c.
[3]

(ii) Find the equation of the curve. [4]

4 A curve has equation y = x2 + 2

x

dy d2 y
(i) Write down expressions for and 2 . [3]
dx dx

(ii) Find the coordinates of the stationary point on the curve and determine its nature. [4]

(iii) Find the volume of the solid formed when the region enclosed by the curve, the x-axis and the
lines x = 1 and x = 2 is rotated completely about the x-axis. [6]
 QUESTIONS 2 TOPIC:8 INTEGRATION

dy
5 A curve is such that = 2x2 − 5. Given that the point (3, 8) lies on the curve, find the equation of
dx
the curve. [4]

4
6 A curve has equation y = √ .
x

(i) The normal to the curve at the point (4, 2) meets the x-axis at P and the y-axis at Q. Find the
length of PQ, correct to 3 significant figures. [6]

(ii) Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and x = 4. [4]

dy 16
7 A curve is such that = , and (1, 4) is a point on the curve.
dx x3

(i) Find the equation of the curve. [4]

(ii) A line with gradient − 12 is a normal to the curve. Find the equation of this normal, giving your
answer in the form ax + by = c. [4]

(iii) Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and x = 2. [4]

dy 4
8 A curve is such that =√ , and P (1, 8) is a point on the curve.
dx (6 − 2x)

(i) The normal to the curve at the point P meets the coordinate axes at Q and at R. Find the
coordinates of the mid-point of QR. [5]

(ii) Find the equation of the curve. [4]

The diagram shows the curve y = x3 − 3x2 − 9x + k, where k is a constant. The curve has a minimum
point on the x-axis.

(i) Find the value of k. [4]

(ii) Find the coordinates of the maximum point of the curve. [1]

(iii) State the set of values of x for which x3 − 3x2 − 9x + k is a decreasing function of x. [1]

(iv) Find the area of the shaded region. [4]

 QUESTIONS 3 TOPIC:8 INTEGRATION

10

The diagram shows the curve y = x(x − 1)(x − 2), which crosses the x-axis at the points O (0, 0),
A (1, 0) and B (2, 0).

(i) The tangents to the curve at the points A and B meet at the point C. Find the x-coordinate of C.
[5]

(ii) Show by integration that the area of the shaded region R1 is the same as the area of the shaded
region R2 . [4]

6
11 The equation of a curve is y = .
5 − 2x

(i) Calculate the gradient of the curve at the point where x = 1. [3]

(ii) A point with coordinates (x, y) moves along the curve in such a way that the rate of increase of
y has a constant value of 0.02 units per second. Find the rate of increase of x when x = 1. [2]

(iii) The region between the curve, the x-axis and the lines x = 0 and x = 1 is rotated through 360◦
about the x-axis. Show that the volume obtained is 12
5
π. [5]

12

1
The diagram shows the curve y = 3x 4 . The shaded region is bounded by the curve, the x-axis and
the lines x = 1 and x = 4. Find the volume of the solid obtained when this shaded region is rotated
completely about the x-axis, giving your answer in terms of π . [4]
 QUESTIONS 4 TOPIC:8 INTEGRATION

8
13 The equation of a curve is y = 2x + .
x2

dy d2 y
(i) Obtain expressions for and 2 . [3]
dx dx

(ii) Find the coordinates of the stationary point on the curve and determine the nature of the stationary
point. [3]

(iii) Show that the normal to the curve at the point (−2, −2) intersects the x-axis at the point (−10, 0).
[3]

(iv) Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and x = 2. [3]

√
14 Find the area of the region enclosed by the curve y = 2 x, the x-axis and the lines x = 1 and x = 4.
[4]

dy
15 A curve is such that = 4 − x and the point P (2, 9) lies on the curve. The normal to the curve at P
dx
meets the curve again at Q. Find
(i) the equation of the curve, [3]
(ii) the equation of the normal to the curve at P, [3]
(iii) the coordinates of Q. [3]
16
y

(1, 18)

(4, 3)

x
O 1 1.6

dy k
The diagram shows a curve for which = − 3 , where k is a constant. The curve passes through the
dx x
points (1, 18) and (4, 3).

16
(i) Show, by integration, that the equation of the curve is y = + 2. [4]
x2
The point P lies on the curve and has x-coordinate 1.6.

(ii) Find the area of the shaded region. [4]

 QUESTIONS 5 TOPIC:8 INTEGRATION

17
y

Q y = Ö (3x + 1)
2

1 P

x
O 1

√
The diagram shows the curve y = (3x + 1) and the points P (0, 1) and Q (1, 2) on the curve. The
shaded region is bounded by the curve, the y-axis and the line y = 2.

(i) Find the area of the shaded region. [4]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. [4]

Tangents are drawn to the curve at the points P and Q.

(iii) Find the acute angle, in degrees correct to 1 decimal place, between the two tangents. [4]

18 The equation of a curve is y = x4 + 4x + 9.

(i) Find the coordinates of the stationary point on the curve and determine its nature. [4]

(ii) Find the area of the region enclosed by the curve, the x-axis and the lines x = 0 and x = 1. [3]

dy 1
19 A curve is such that = 3x 2 − 6 and the point (9, 2) lies on the curve.
dx
(i) Find the equation of the curve. [4]

(ii) Find the x-coordinate of the stationary point on the curve and determine the nature of the
stationary point. [3]
 QUESTIONS 6 TOPIC:8 INTEGRATION

20
y

a
y= x

x
O 1 3

a
The diagram shows part of the curve y = , where a is a positive constant. Given that the volume
x
obtained when the shaded region is rotated through 360◦ about the x-axis is 24π , find the value of a.
[4]

21
y

2
y = (x – 2)
A

y + 2x = 7

B
x
O

The diagram shows the curve y = (x − 2)2 and the line y + 2x = 7, which intersect at points A and B.
Find the area of the shaded region. [8]

dy 6
22 The equation of a curve is such that = √ . Given that the curve passes through the point
dx (3x − 2)
P (2, 11), find
(i) the equation of the normal to the curve at P, [3]
(ii) the equation of the curve. [4]
 QUESTIONS 7 TOPIC:8 INTEGRATION

23
y

y = x + 4x

y=5
A B
M

x
O

4
The diagram shows part of the curve y = x + which has a minimum point at M . The line y = 5
x
intersects the curve at the points A and B.

(i) Find the coordinates of A, B and M . [5]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. [6]

The equation of a curve is y =

9
24
2−x
.

dy
(i) Find an expression for and determine, with a reason, whether the curve has any stationary
dx
points. [3]

(ii) Find the volume obtained when the region bounded by the curve, the coordinate axes and the
line x = 1 is rotated through 360◦ about the x-axis. [4]

(iii) Find the set of values of k for which the line y = x + k intersects the curve at two distinct points.
[4]
25 y

x=5
A

1
y=
B 1
(3x + 1)4

x
O

1
The diagram shows part of the curve y = 1
. The curve cuts the y-axis at A and the line x = 5
(3x + 1) 4
at B.

(i) Show that the equation of the line AB is y = − 110 x + 1. [4]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. [9]
 QUESTIONS 8 TOPIC:8 INTEGRATION

26 A curve has equation y = f (x). It is given that f ′ (x) = 3x2 + 2x − 5.

(i) Find the set of values of x for which f is an increasing function. [3]

(ii) Given that the curve passes through (1, 3), find f (x). [4]

27
y

y = 9 – x3

Q 8
y=
x3
x
O a b

8
The diagram shows parts of the curves y = 9 − x3 and y = and their points of intersection P and Q.
x3
The x-coordinates of P and Q are a and b respectively.

(i) Show that x = a and x = b are roots of the equation x6 − 9x3 + 8 = 0. Solve this equation and
hence state the value of a and the value of b. [4]

(ii) Find the area of the shaded region between the two curves. [5]

(iii) The tangents to the two curves at x = c (where a < c < b) are parallel to each other. Find the
value of c. [4]

28 (i) Sketch the curve y = (x − 2)2 . [1]

(ii) The region enclosed by the curve, the x-axis and the y-axis is rotated through 360◦ about the
x-axis. Find the volume obtained, giving your answer in terms of π . [4]

dy 3
29 A curve is such that = and the point (1, 21 ) lies on the curve.
dx (1 + 2x)2

(i) Find the equation of the curve. [4]

(ii) Find the set of values of x for which the gradient of the curve is less than 13 . [3]
 QUESTIONS 9 TOPIC:8 INTEGRATION

Find ä x3 +  dx.

1
30 [3]
x3

31
y

M
y = 4 Öx – x

x
O A

√
The diagram shows part of the curve y = 4 x − x. The curve has a maximum point at M and meets
the x-axis at O and A.

(i) Find the coordinates of A and M . [5]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis, giving
your answer in terms of π . [6]

2x3 + 5
32 (a) Differentiate with respect to x. [3]
x
1
(b) Find ã (3x − 2)5 dx and hence find the value of ã (3x − 2)5 dx. [4]
0

dy 2
33 A curve is such that = √ − 1 and P (9, 5) is a point on the curve.
dx x

(i) Find the equation of the curve. [4]

(ii) Find the coordinates of the stationary point on the curve. [3]

d2 y
(iii) Find an expression for and determine the nature of the stationary point. [2]
dx2

(iv) The normal to the curve at P makes an angle of tan−1 k with the positive x-axis. Find the value
of k. [2]

34 A function f is defined for x ∈ > and is such that f ′ (x) = 2x − 6. The range of the function is given by
f (x) ≥ −4.

(i) State the value of x for which f (x) has a stationary value. [1]

(ii) Find an expression for f (x) in terms of x. [4]

 QUESTIONS 10 TOPIC:8 INTEGRATION

35
y
y = Ö(1 + 2x)
C

x
A O

√
The diagram shows the curve y = (1 + 2x) meeting the x-axis at A and the y-axis at B. The
y-coordinate of the point C on the curve is 3.

(i) Find the coordinates of B and C. [2]

(ii) Find the equation of the normal to the curve at C. [4]

(iii) Find the volume obtained when the shaded region is rotated through 360◦ about the y-axis. [5]

dy 8
36 A curve is such that = 5 − 2 . The line 3y + x = 17 is the normal to the curve at the point P on the
dx x
curve. Given that the x-coordinate of P is positive, find
(i) the coordinates of P, [4]
(ii) the equation of the curve. [4]

√
37 The equation of a curve is y = (8x − x2 ). Find
dy
(i) an expression for , and the coordinates of the stationary point on the curve, [4]
dx
(ii) the volume obtained when the region bounded by the curve and the x-axis is rotated through
360◦ about the x-axis. [4]

38 A curve y = f (x) has a stationary point at P (3, −10). It is given that f ′ (x) = 2x2 + kx − 12, where k is
a constant.

(i) Show that k = −2 and hence find the x-coordinate of the other stationary point, Q. [4]

(ii) Find f ′′ (x) and determine the nature of each of the stationary points P and Q. [2]

(iii) Find f (x). [4]

 QUESTIONS 11 TOPIC:8 INTEGRATION

39
y

y = Ö(x + 1)

y=x+1

–1 x
O

√
The diagram shows the line y = x + 1 and the curve y = (x + 1), meeting at (−1, 0) and (0, 1).

(i) Find the area of the shaded region. [5]

(ii) Find the volume obtained when the shaded region is rotated through 360◦ about the y-axis. [7]

40
y

2
y=
Ö(x + 1)

y=1

x
O

The diagram shows the line y = 1 and part of the curve y = √

2
(x + 1)
.

(i) Show that the equation y = √ can be written in the form x = 2 − 1.

2 4
(x + 1)
[1]
y

(ii) Find ä  − 1 dy. Hence find the area of the shaded region.
4
[5]
y2

(iii) The shaded region is rotated through 360◦ about the y-axis. Find the exact value of the volume
of revolution obtained. [5]
 QUESTIONS 12 TOPIC:8 INTEGRATION

41
y

y= 6
2x – 3

x
O 2 3

6
The diagram shows the region enclosed by the curve y = , the x-axis and the lines x = 2 and
2x − 3
x = 3. Find, in terms of π , the volume obtained when this region is rotated through 360◦ about the
x-axis. [4]

42
y
A

y = –x2 + 8x –10

x
O

The diagram shows part of the curve y = −x2 + 8x − 10 which passes through the points A and B. The
curve has a maximum point at A and the gradient of the line BA is 2.

(i) Find the coordinates of A and B. [7]

(ii) Find ã y dx and hence evaluate the area of the shaded region. [4]
 QUESTIONS 13 TOPIC:8 INTEGRATION

43
y
A

1 B (6, 1) 8
x= –2
y2

x
O

8
The diagram shows part of the curve x = − 2, crossing the y-axis at the point A. The point B (6, 1)
y2
lies on the curve. The shaded region is bounded by the curve, the y-axis and the line y = 1. Find the
exact volume obtained when this shaded region is rotated through 360◦ about the y-axis. [6]

d2 y
44 A curve is such that = −4x. The curve has a maximum point at (2, 12).
dx2
(i) Find the equation of the curve. [6]

A point P moves along the curve in such a way that the x-coordinate is increasing at 0.05 units
per second.

(ii) Find the rate at which the y-coordinate is changing when x = 3, stating whether the y-coordinate
is increasing or decreasing. [2]

dy 8
45 A curve is such that = − 3 − 1 and the point (2, 4) lies on the curve. Find the equation of the
dx x
curve. [4]

46
y
3y = 2x – 1

y2 = 2x – 1

O x
1 a
2

The diagram shows the curve y2 = 2x − 1 and the straight line 3y = 2x − 1. The curve and straight line
intersect at x = 12 and x = a, where a is a constant.

(i) Show that a = 5. [2]

(ii) Find, showing all necessary working, the area of the shaded region. [6]
 QUESTIONS 14 TOPIC:8 INTEGRATION

47
y

B (0, 3)

y= 9
2x + 3
C A (3, 1)

x
O

9
The diagram shows part of the curve y = , crossing the y-axis at the point B (0, 3). The point
2x + 3
A on the curve has coordinates (3, 1) and the tangent to the curve at A crosses the y-axis at C.

(i) Find the equation of the tangent to the curve at A. [4]

(ii) Determine, showing all necessary working, whether C is nearer to B or to O. [1]

(iii) Find, showing all necessary working, the exact volume obtained when the shaded region is
rotated through 360◦ about the x-axis. [4]

dy 4
48 A curve is defined for x > 0 and is such that = x + 2 . The point P (4, 8) lies on the curve.
dx x
(i) Find the equation of the curve. [4]

(ii) Show that the gradient of the curve has a minimum value when x = 2 and state this minimum
value. [4]

49 A curve is such that

dy 3
= 2(3x + 4) 2 − 6x − 8.
dx

d2 y
(i) Find . [2]
dx2
(ii) Verify that the curve has a stationary point when x = −1 and determine its nature. [2]

(iii) It is now given that the stationary point on the curve has coordinates (−1, 5). Find the equation
of the curve. [5]
 QUESTIONS 15 TOPIC:8 INTEGRATION

50
y

2
y = x(x – 2)

O x
b a

The diagram shows the curve with equation y = x(x − 2)2 . The minimum point on the curve has
coordinates (a, 0) and the x-coordinate of the maximum point is b, where a and b are constants.

(i) State the value of a. [1]

(ii) Find the value of b. [4]

(iii) Find the area of the shaded region. [4]

dy
(iv) The gradient, , of the curve has a minimum value m. Find the value of m. [4]
dx

1 − 12
51 A curve has equation y = f x
and is such that f ′ x
= 3x 2 + 3x − 10.
1
(i) By using the substitution u = x 2 , or otherwise, find the values of x for which the curve y = f x

has stationary points. [4]

(ii) Find f ′′ x
and hence, or otherwise, determine the nature of each stationary point. [3]

(iii) It is given that the curve y = f x
passes through the point 4, −7
. Find f x
. [4]

52
y

C A (1, 1) y = (x – 2)4

x
O B

The diagram shows part of the curve y = x − 2
4 and the point A 1, 1
on the curve. The tangent at
A cuts the x-axis at B and the normal at A cuts the y-axis at C.

(i) Find the coordinates of B and C. [6]


a
(ii) Find the distance AC, giving your answer in the form , where a and b are integers. [2]
b
(iii) Find the area of the shaded region. [4]
 QUESTIONS 16 TOPIC:8 INTEGRATION

dy 6
53 A curve is such that = 2 and 2, 9
is a point on the curve. Find the equation of the curve. [3]
dx x

54
y
y = Ö(1 + 4x)

x
A O C


The diagram shows the curve y = 1 + 4x
, which intersects the x-axis at A and the y-axis at B. The
normal to the curve at B meets the x-axis at C. Find
(i) the equation of BC, [5]
(ii) the area of the shaded region. [5]

dy 
55 A curve is such that = 2x + 5
and 2, 5
is a point on the curve. Find the equation of the curve.
dx
[4]

56
y

A (1, 7)

8
y= –x
Öx
C

B (4, 0) x
O

8
The diagram shows part of the curve y =  − x and points A 1, 7
and B 4, 0
which lie on the
x
curve. The tangent to the curve at B intersects the line x = 1 at the point C.

(i) Find the coordinates of C. [4]

(ii) Find the area of the shaded region. [5]

 QUESTIONS 17 TOPIC:8 INTEGRATION

1 6
57 A curve has equation y = f x
. It is given that f ′ x
=  + 2 and that f 3
= 1. Find f x
. [5]
x + 6
x

58
y

y = (3 – 2x)3

(12 , 8)

x
O
1 
The diagram shows the curve y = 3 − 2x
3 and the tangent to the curve at the point 2
,8 .

(i) Find the equation of this tangent, giving your answer in the form y = mx + c. [5]

(ii) Find the area of the shaded region. [6]

The equation of a curve is y = 

2
59
5x − 6

.

(i) Find the gradient of the curve at the point where x = 2. [3]
3

(ii) Find Ô  dx and hence evaluate Ô 

2 2
dx.
5x − 6
5x − 6

[4]
2

60
y
C y = 8x + 2x
A
B

x
O
 QUESTIONS 18 TOPIC:8 INTEGRATION

The diagram shows part of the curve y = + 2x and three points A, B and C on the curve with
8
x
x-coordinates 1, 2 and 5 respectively.

(i) A point P moves along the curve in such a way that its x-coordinate increases at a constant rate
of 0.04 units per second. Find the rate at which the y-coordinate of P is changing as P passes
through A. [4]

(ii) Find the volume obtained when the shaded region is rotated through 360Å about the x-axis. [6]

− 32
61 A curve has equation y = f x
. It is given that f ′ x
= x + 1 and that f 4
= 5. Find f x
. [4]

62
y

y = Ö(x 4 + 4x + 4)

x
–1 O

The diagram shows the curve y = x4 + 4x + 4
.

(i) Find the equation of the tangent to the curve at the point 0, 2
. [4]

(ii) Show that the x-coordinates of the points of intersection of the line y = x + 2 and the curve are
given by the equation x + 2
2 = x4 + 4x + 4. Hence find these x-coordinates. [4]

(iii) The region shaded in the diagram is rotated through 360Å about the x-axis. Find the volume of
revolution. [4]

dy 1 −1  
63 A curve is such that = x 2 − x 2 . The curve passes through the point 4, 23 .
dx
(i) Find the equation of the curve. [4]

d2 y
(ii) Find . [2]
dx2
(iii) Find the coordinates of the stationary point and determine its nature. [5]
 QUESTIONS 19 TOPIC:8 INTEGRATION

d2 y
64 The equation of a curve is such that = 2x − 1. Given that the curve has a minimum point at
dx2
3, −10
, find the coordinates of the maximum point. [8]

65
y

y = 8 − ï 4 − x

P 3, 7

x
O

The diagram shows part of the curve y = 8 − 4 − x
and the tangent to the curve at P 3, 7
.

dy
(i) Find expressions for and Ó y dx. [5]
dx

(ii) Find the equation of the tangent to the curve at P in the form y = mx + c. [2]

(iii) Find, showing all necessary working, the area of the shaded region. [4]

dy 12
66 A curve is such that =  , where a is a constant. The point P 2, 14
lies on the curve and
dx 4x + a

the normal to the curve at P is 3y + x = 5.

(i) Show that a = 8. [3]

(ii) Find the equation of the curve. [4]

 QUESTIONS 20 TOPIC:8 INTEGRATION

67
y

y = 2x + 1

y = −x2 + 12x − 20

x
O

The diagram shows the curve y = −x2 + 12x − 20 and the line y = 2x + 1. Find, showing all necessary
working, the area of the shaded region. [8]