Education Resources
Integration
Higher Mathematics Supplementary Resources
Section A
This section is designed to provide examples which develop routine skills necessary for
completion of this section.
R1
I can evaluate the definite integral of a polynomial functions with integer
limits.
1.
Find
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
2.
Find
Higher Mathematics – Integration
Page 1
R2
I can evaluate the definite integral of a function with limits in radians, surds
or fractions.
1.
Evaluate
(a)
(b)
(d)
2.
(e)
(f)
Evaluate
(a)
(b)
(c)
(d)
3.
(c)
Evaluate
(a)
(b)
(c)
(d)
(e)
(f)
R3
I can apply a standard integral of the form
1.
Find
with
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
SLC Education Resources – Biggar High School
.
Page 2
2.
Find
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
3.
Find
R4
I can integrate
1.
Find
and
(a)
by first making a substitution.
(b)
(c)
(d)
Higher Mathematics – Integration
Page 3
Section B
This section is designed to provide examples which develop Course Assessment level
skills
NR1
I can evaluate one of the limits of a definite integral given the value of the
definite integral.
1.
Find a, when
(a)
2.
Given that,
3.
Find a for
(b)
calculate the value of .
given:
(a)
4.
5.
6.
Given that
(b)
,
, calculate the value of .
Determine , given that
Given that
SLC Education Resources – Biggar High School
, find .
Page 4
NR2
I can evaluate the area enclosed between a function and the x axis.
1.
Find the shaded area in the following diagrams
(a)
(b)
y
y  5x  x
2
y
y  4 x  x2
x
x
y
(c)
y  3  3x 2
x
2.
y
(d)
-1
y  10  4 x  3x2
1
x
The diagram shows part of the graph
of
.
(a) Find the coordinates of P
and Q.
(b) Find the shaded area.
3.
The diagram shows part of
the graph of
–
(a)
Find the coordinates of
P and Q.
(b)
Calculate the shaded
area.
Higher Mathematics – Integration
Page 5
4.
5.
The diagram shows the graph
of
–
–
(a)
Find the coordinates of A
and B.
(b)
Calculate the shaded
area.
Which of the following gives the numerical value of the shaded area?
(I)

(II)

1
1
0
1
(III) 2
A (I) only
6.
x dx
x dx 


1
x dx
0
1
x dx
0
B (II) only
C (III) only
D (I), (II) and (III)
An artist has designed a
“bow” shape which he
finds can be modelled
by the shaded area
shown.
Calculate the area of
this shape.
SLC Education Resources – Biggar High School
O
Page 6
7.
The graph below has equation
.
The total shaded aread is
bounded by the curve, the -axis,
the -axis and the line
.
8.
(a)
Calculate the shaded area
labelled S.
(b)
Hence find the total shaded
area.
Diagram 1 shows the profit/loss
function for the manufacture of
thousand kitchen units.
y
1
y  x2  x
3
diagram 1
The profit/loss is measured in millions
of £s and is represented by the area
between the function and the -axis .
0
x
h
Any area below the -axis represents a
loss; any area above the -axis
represents a profit.
1
The profit/loss function is given by f ( x)  x 2  x where x  0 .
3
(a)
Find the value of .
y
(b)
Diagram 2 (not drawn to
scale) represents the
breakeven situation where
the initial loss made on
selling the first h thousand
units is exactly balanced by
the later profit.
diagram 2
0
x
h
k
Calculate the value of .
Higher Mathematics – Integration
Page 7
9.
The fire surround is rectangular, with a parabolic opening for the fire.
(a)
Find the coordinates of A and B.
(b)
Calculate the tiled area (1 unit = 10cm).
120cm
160cm
A
10.
O
B
Millennium Park is based on two parabola shapes.
A plan is laid out in the coordinate diagram.
Calculate the areas of the three parts of the park on the diagram.
picnic area
animal farm
gardens
SLC Education Resources – Biggar High School
Page 8
NR3
I can evaluate the area enclosed between two functions.
1.
2.
3.
4.
Calculate the area enclosed by the line
Higher Mathematics – Integration
and the parabola
Page 9
5.
This diagram shows a rough sketch of the quadratic
function y  6 x  x 2 .The tangent at the maximum
stationary point has been drawn.
(a) Explain clearly why the tangent has equation y = 9.
y
x
O
(b) Calculate the shaded area enclosed by the curve,
the tangent and the y-axis.
6.
The diagram opposite shows the curve
and the line
(a) Find the coordinates of A and B.
(b) Calculate the shaded area.
7.
The curves with equations
and
–
intersect at K and L.
Calculate the area enclosed by these two
curves.
8.
The curve
line
–
–
–
and the
are shown opposite.
(a) B has coordinates (1,-2). Find the
coordinates of A and C.
(b) Hence calculate the shaded area.
SLC Education Resources – Biggar High School
Page 10
9.
The diagram opposite shows an area
enclosed by 3 curves:
and
(a) P and Q have coordinates (p,4) and (q,1).
Find the values of p and q.
(b) Calculate the shaded area.
10.
This diagram shows 2 curves y  f1 ( x) and y  f 2 ( x) which intersect at x  a .
The area of the shaded region is
A
B
C
D




b
f1 ( x)dx 
0
a
f 2 ( x)dx 
0
a
f 2 ( x)dx 
0
a
f1 ( x)dx 
0

a
f 2 ( x)dx
0



b
f1( x)dx
a
b
f1( x)dx
a
b
f 2 ( x)dx
a
11.
Higher Mathematics – Integration
Page 11
12.
13.
SLC Education Resources – Biggar High School
Page 12
14.
15.
16. The diagram shows the curves y  sin x
and y  cos x for 0  x   2 .
The shaded area is given by

A
B

 4
cos x dx 
 2
 4
0
 4
0

sin x dx 

cos x dx
 2
 4
sin x dx
Higher Mathematics – Integration
C

D
 2
(sin x  cos x) dx
0

 2
(sin x  cos x) dx
0
Page 13
17.
18.
The diagram shows the front of a packet of Vitago,
a new vitamin preparation to provide early morning
energy. The shaded region is red and the rest yellow.
The design was created by drawing the
curves y  9 x 2 and y  4 x 2  2 x3 as shown in the diagram below.
The edges of the packet are represented by the coordinate axes and the lines x  43 and y  4 .
Show that
of the front of the packet is red.
SLC Education Resources – Biggar High School
Page 14
19. a) Find the x-values of the three points of intersection for these curves,
for
.
b) Calculate the area enclosed by the curves.
1
-1
-
Higher Mathematics – Integration
Page 15
NR4
I can solve differential equations of the form
and give a particular
solution.
1.
Given the gradient
of the curve at the point
and a point on the curve,
find the equation of each curve:
2.
a)
(3,4)
b)
(1,9)
Find the solution to the following differential equations:
a)
b)
3.
and
and
A curve has gradient given by
= . The curve passes through the point (9,10).
Find the equation of the curve.
4.
5.
6.
7.
SLC Education Resources – Biggar High School
Page 16
NR5
I have experience of cross topic exam standard questions.
Integration and the addition formula
1. a)
Write down a formula for
for
b)
, and use it to solve the equation
.
Find the shaded area enclosed by the curves
and
.
1
O
Integration and quadratic graphs
2.
The parabola shown crosses the axis at
and
, and has a
maximum at
.
The shaded area is bound by the
parabola, the -axis and the lines
and
.
(a)
Find the equation of the
parabola.
(b)
Hence show that the shaded
area, , is given by
.
Higher Mathematics – Integration
Page 17
Integration and the wave function
3.
(a)
The expression
where
and
Calculate the values of
(b)
can be written in the form
.
and .
Hence find the value of , where
, for which
Integration and Differentiation
4.
(a)
Find the equation of the tangent to the parabola
at P(2,1)
(b)
Calculate the area of the shaded region bounded by the tangent, the
parabola and the y axis.
P(2,1)
O
SLC Education Resources – Biggar High School
Page 18
Integration and polynomials
5.
(a)
(i)
Show that
(ii)
Factorise
is a factor of
(iii) Solve
(b)
.
fully.
.
The diagram shows the curve with equation
The curve crosses the -axis at ,
and .
Determine the shade area.
Higher Mathematics – Integration
Page 19
Answers
R1
1. (a)
2. (a)
(b)
0
(g)
(b)
(c)
6
(h)
(c)
(d)
(e)
(f)
-6
(d)
(e)
(f)
0
(d)
(e)
0
(f)
(e)
3+4
(f)
-15
(i)
R2
1. (a)
0
(b)
2. (a)
2
(b)
(c)
(d)
(b)
(c)
(d)
3. (a)
0
(c)
0
R3
1. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
2. (a)
(b)
(c)
(d)
(e)
(f)
3. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
SLC Education Resources – Biggar High School
Page 20
R4
1. (a)
(b)
(c)
(d)
NR1
1(a)
1(b)
2.
3(a)
3(b)
4.
5.
6.
NR2
1(a)
units2
1(b)
units2
1(c)
2(a)
2(b)
3(a)
3(b)
4(a)
4(b)
5.
6.
8(a)
8(b)
9(b)
units2
7(a)
units2
1(d)
units2
units2
7(b)
units2
units2
4.
units2
6(b)
units2
units2
units2
units2
9(a)
cm2
10. (a)
(b)
(c)
NR3
1.
units2
5(a).
7.
5(b)
units2
9(a)
12.
2.
units2
3.
6(a)
8(a)
9(b)
units2
units2
units2
13.
14(b)
Higher Mathematics – Integration
units2
10.
8(b)
units2
11.
units2
14(a)
14(c)
units2
Page 21
15(a)
17.
and
units2
15(b)
units2
16.
18.
19(a)
19(b)
units2
NR4
1(a)
1(b)
2(a)
2(b)
3.
4.
5.
6.
7.
NR5
1(a)
1(b)
2(a)
2(b)
units2
3(a)
3(b)
4(a)
4(b)
units2
5(a)i remainder = 0 therefore
is a factor
5(a)ii
5(b)
5(a)iii
10 units2
SLC Education Resources – Biggar High School
Page 22