Maths Quest Maths B Year 12 for Queensland
Chapter 7 Introduction to integration WorkSHEET 7.1
WorkSHEET 7.1 Introduction to integration
1
2
Name: ___________________________
Find an approximation for the area between the
1 2
curve and the x-axis over the interval indicated y  4 x Relevant points on graph are:
in the diagram below using:
1
9
(0, 0), (1, ), (2, 1), (3, ), (4, 4)
(a) the upper rectangles.
4
4
(b)
the lower rectangles.
(c)
the average of the upper and lower
rectangles.
1
(a)
Total area of upper rectangles is
1
9
1  1 1   1  4 1
4
4
1
 7 square units.
2
(b)
Total area of lower rectangles is
1
9
0 1  1  11  1
4
4
1
 3 square units.
2
(c)
Average of the two areas is
1
1
1
(7  3 )  2  5 square units.
2
2
2
Find an approximation for the area between the
1 3
curve and the x-axis over the interval indicated y  3 x Relevant points on graph are:
in the diagram below using the area of the
1
8
(0, 0), (1, ), (2, ), (3, 9)
trapezium.
3
3
Area of trapezium is
1
1
1 1 8
1 8
 (0  )  1   (  )  1   (  9)  1
2
3
2 3 3
2 3
1
 7 square units.
2
Maths Quest Maths B Year 12 for Queensland
3
Chapter 7 Introduction to integration WorkSHEET 7.1
Find an approximation for the area under the
4
y 2
4
x
graph y  2 and the x-axis over the interval
x
x = 0.5 to x = 3.5 using the trapezoidal rule and: (a) Relevant points on graph are:
(a) interval widths of 1 unit
16
(0.5, 16), (1.5, ), (2.5, 0.64),
9
(b) interval widths of 0.5 units
16
),
(3.5,
49
Area of trapezium is
1
16
1 16
 (16  )  1   (  0.64)  1 
2
9
2
9
1
16
 (0.64  )  1  10.6 square units.
2
49
(b)
4
Antidifferentiate the following functions:
(a) 2x2
(b)
Relevant points on graph are:
16
(0.5, 16), (1, 4), (1.5, ), (2, 1),
9
4
16
),
(2.5, 0.64), (3, ), (3.5,
9
49
Area of trapezium is
1
1 1
16 1
 (16  4)    (4  )  
2
2 2
9
2
1 16
1 1
1
 (  1)    (1  0.64)  
2
9
2 2
2
1
4 1 1 4 16 1
 (0.64  )    (  ) 
2
9 2 2 9 49 2
 8.0 square units.
dy
a
 ax n , then y 
x n1  c.
n  1
dx
dy
 2x 2
dx
2
y  x3  c
3
(a)
If
(b)
y
3x4
3 5
x c
5
2
Maths Quest Maths B Year 12 for Queensland
5
Chapter 7 Introduction to integration WorkSHEET 7.1
Antidifferentiate the following functions:
9
(a)
x2
(b)
(a)
x
dy
9
 2
dx x
dy
 9 x 2
dx
y

(b)
9
 2  1x 21
9
c
x
dy
 x
dx
1
dy
 x2
dx
1
1 12
y
x
c
1  12

y
6
Antidifferentiate the following functions:
(a) x x
(b)
(a)
(x + 2)3
(b)
7
If f x   3x 2  4 x and f 1  4 find an
expression for f x  .

2 32
x c
3
dy
x x
dx
3
dy
 x2
dx
2 5
y  x2 c
5
dy
  x  2 3
dx
1
y   x  2 4
4
If f  x   3x 2  4 x,
then f x   x 3  2 x 2  c.
If f 1  4, then
13  2  12  c  4
c5
That is, f x   x 3  2 x 2  5
3
Maths Quest Maths B Year 12 for Queensland
8
Chapter 7 Introduction to integration WorkSHEET 7.1
Find the following indefinite integrals:
(a)   x  2  2  x  3 dx
(b)
 4  3x 
3
(a)
dx
 x  2 x  3 dx
  x 2  4 x  4x  3 dx
  x 3  x 2  8 x  12 dx
2

(b)
 4  3x 


9
Antidifferentiate the following:
4
dx
(a) 
3x  2
(b)
(a)
1 4 1 3
x  x  4 x 2  12 x  c
4
3
3
dx
4  3x 2
 2  3
1
6 4  3 x  2
c
4
 3x  2 dx
1
 4  log e 3 x  2   c
3
4
 log e 3 x  2   c
3
5  2 x  x3
dx

x
(b)

5  2x  x3
  5x
x

1
 10 x 2
1
2
dx
1
 2x 2
3

5
x2
dx
7
4
2
 x2  x2  c
3
7
4
Maths Quest Maths B Year 12 for Queensland
10
Chapter 7 Introduction to integration WorkSHEET 7.1
A curve has the gradient, g x   k x  3x 3
where k is a constant and a stationary point
(1, 2). Find:
(a) the value of k
(b)
(c)
(a)
g x   k x  3x 3
when x = 1, gradient = 0
(b)
0  k 1  3  13
3k
g(x)
g x  
g(4)
1
3x 2
 3x 3
3
2
3
g x    3x 2  x 4  c
3
4
when x = 1, g(x) = 2
3
(c)
2
3
 2   3  1 2   14  c
3
4
3
2  2 c
4
1
c  3
4
g x  
3
2x 2
g 4  
3
2 42

3 4 13
x 
4
4
3
13
  44 
4
4
1
 16  192  3
4
1
 179
4
5