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VCE 12 Methods 2023

7.08 The definite integral and areas under curves

Worksheet
Definite integrals
1

Evaluate the following definite integrals:

a

\int_{0}^{3} \left( 4 x + 5\right) dx

b

\int_{ - 2 }^{0} \left( 10 x + 4\right) dx

c

\int_{2}^{4} \left( 6 x + 5\right) dx

d

\int_{3}^{4} \left( - 4 x + 3\right) dx

e

\int_{ - 4 }^{5} \left( - 8 x + 3\right) dx

f

\int_{ - 10 }^{ - 6 } \left( - 5 x + 8\right) dx

g

\int_{ - 2 }^{0} 9 x^{2} dx

h

\int_{ - 1 }^{3} 9 x^{2} dx

i

\int_{ - 2 }^{1} \left(x^{2} + 4\right) dx

j

\int_{ - 1 }^{2} \left( 9 x^{2} + 1\right) dx

k

\int_{4}^{6} \left( 9 x^{2} + 2 x + 7\right) dx

l

\int_{ - 4 }^{2} x \left(x - 4\right) dx

m

\int_{0}^{4} 5 x^{\frac{3}{2}} dx

n

\int_{ - 3 }^{6} x \left(x + 3\right) \left(x - 6\right) dx

o

\int_{6}^{11} \sqrt{x - 2} dx

p

\int_{3}^{6} \left(\sqrt{x - 2} + 5\right) dx

q

\int_{3}^{4} \left( 2 x + 3\right)^{3} dx

r

\int_{ - 4 }^{2} \left(\left(x + 2\right)^{3} + 3\right) dx

s

\int_{1}^{2} \dfrac{x^{5} - x^{ - 2 }}{x^{2}} dx

Properties of definite integrals
2

Consider the function f \left( x \right) = x^2. Find the value of the following:

a
\int_{1}^{1} f \left( x \right) dx.
b
\int_{5}^{5} f \left( x \right) dx.
c
\int_{-2}^{-2} f \left( x \right) dx.
d
\int_{a}^{a} f \left( x \right) dx.
3

Consider the function f \left( x \right) = 2 x.

a

Find the value of \int_{0}^{3} f \left( x \right) dx.

b

Find the value of \int_{0}^{3} 5f \left( x \right) dx.

c

Find the value of 5\int_{0}^{3} f \left( x \right)dx.

d

State the property of definite integrals demonstrated by parts (b) and (c).

4

Given that \int_{ - 2 }^{7} f \left( x \right) dx = 2, find \int_{ - 2 }^{7} 5 f \left( x \right) dx.

5

Given that \int_{ - 1 }^{6} f \left( x \right) dx = 3, find \int_{ - 1 }^{6} \left( 9 f \left( x \right) - 2\right) dx.

6

Consider the function f \left( x \right) = 6 x.

a

Find the value of \int_{4}^{8} f \left( x \right) dx.

b

Find the value of \int_{8}^{4} f \left( x \right) dx.

c

State the property of definite integrals demonstrated by parts (a) and (b).

7

Given that \int_{4}^{6} f \left( x \right) dx = 3, find the values of the following:

a

\int_{6}^{4} f \left( x \right) dx

b

\int_{4}^{6} 3 f \left( x \right) dx

c

\int_{4}^{6} \left(f \left( x \right) + x\right) dx

8

Given that \int_{ - 1 }^{2} f \left( x \right) dx = 4 and \int_{2}^{8} f \left( x \right) dx = 8, find the values of the folowing:

a

\int_{ - 1 }^{8} f \left( x \right) dx

b

\int_{8}^{ - 1 } f \left( x \right) dx

c

\int_{ - 1 }^{2} 2 f \left( x \right) dx + \int_{2}^{8} 3 f \left( x \right) dx

9

Given that \int_{ - 1 }^{3} f \left( x \right) dx = 5 and \int_{2}^{3} f \left( x \right) dx = 2, find the values of the following:

a

\int_{ - 1 }^{2} f \left( x \right) dx

b

\int_{3}^{ - 1 } f \left( x \right) dx

c

2 \int_{ - 1 }^{2} f \left( x \right) dx + \int_{2}^{3} 3 f \left( x \right) dx

Signed areas
10

Consider the graph of y = f \left( x \right).

a

Find the value of \int_{0}^{2} f \left( x \right) dx.

b

Find the value of \int_{2}^{5} f \left( x \right) dx.

c

Find the value of \int_{5}^{6} f \left( x \right) dx.

d

Hence state the area bounded by the function and the x-axis.

e

Write a single definite integral to represent the area bounded by the function and the x-axis.

1
2
3
4
5
6
x
1
2
3
4
5
6
y
11

Consider the graph of y = f \left( x \right).

a

Find the value of \int_{0}^{4} f \left( x \right) dx.

b

Find the value of \int_{4}^{6} f \left( x \right) dx.

c

Find the value of \int_{6}^{8} f \left( x \right) dx.

d

Hence state the area bounded by the function and the x-axis.

e

Write a single definite integral to represent the area bounded by the function and the x-axis.

1
2
3
4
5
6
7
8
x
1
2
3
4
5
6
7
8
y
12

Consider the graph of y = f \left( x \right).

a

Find the value of \int_{0}^{5} f \left( x \right) dx.

b

Find the value of \int_{5}^{7} f \left( x \right) dx.

c

Hence find the value of \int_{0}^{7} f \left( x \right) dx.

d

State the area bounded by the function and the x-axis.

e

Write a single definite integral to represent the area bounded by the function and the x-axis.

1
2
3
4
5
6
7
x
-7
-6
-5
-4
-3
-2
-1
y
13

Consider the graph of y = f \left( x \right):

a

Find the value of \int_{0}^{4} f \left( x \right) dx.

b

Find the exact value of \int_{4}^{16} f \left( x \right) dx.

c

Hence find the exact value of \int_{0}^{16} f \left( x \right) dx.

d

Calculate the exact area bounded by the curve, the x-axis and the y-axis.

e

Write an expression for the exact area from part (d) as a sum or difference of definite integrals.

2
4
6
8
10
12
14
16
x
-6
-4
-2
2
4
6
y
14

Consider the function y = -5.

a

State whether the graph is above or below the x-axis.

b

Calculate \int_{ - 4 }^{3} \left(-5\right) \ dx.

c

Hence find the area bounded by the curve, the x-axis and the bounds x = - 4 and x = 3.

15

Consider the function y = 2 x - 8.

a

Find the x-intercept of the function.

b

State the values of x for which the graph is above the x-axis.

c

Calculate \int_{ - 2 }^{4} \left( 2 x - 8\right) dx.

d

Hence find the area bounded by the line, the x-axis and the bounds x = - 2 and x = 4.

16

Consider the function y = \left(x - 3\right) \left(x - 9\right).

a

Find the x-intercepts of the function.

b

State the values of x for which the graph is below the x-axis.

c

Calculate \int_{3}^{9} \left(x - 3\right) \left(x - 9\right) dx.

d

Hence find the area bounded by the curve, the x-axis and the bounds x = 3 and x = 9.

17

Consider the function y = - \left(x + 2\right) \left(x + 8\right).

a

Find the x-intercepts of the function.

b

State the values of x for which the graph is above the x-axis.

c

Calculate \int_{ - 8 }^{ - 2 } - \left(x + 2\right) \left(x + 8\right) dx.

d

Hence find the area bounded by the curve, the x-axis and the bounds x = - 8 and \\x = - 2.

Area under curve
18

Consider the graph of the curve y = x^{2} + 6.

Find the exact area of the shaded region.

19

Consider the graph of the curve y = 6 x^{2}.

Find the exact area of the shaded region.

20

Consider the graph of the line x + y = 3.

Find the exact area of the shaded region.

1
2
3
4
5
6
7
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
21

Consider the graph of the curve y = 4 - x^{2}.

Find the exact area of the shaded region.

22

Consider the graph of the curve \\y = x \left(x - 1\right) \left(x + 3\right).

Find the exact area of the shaded region.

23

Consider the given functions.

i

Sketch a graph of the function.

ii

Hence calculate the exact area bounded by the curve, x = 1, x = 3 and the x-axis.

a

y = 2 x + 3

b

y = - 2 x + 8

c

y = 4x- x^2 - 3

24

Consider the function y = \left(x - 1\right)^{2} \left(x + 3\right).

a

Sketch a graph of the function.

b

Hence calculate the exact area bounded by the curve, x-axis and the lines x = - 1 and x = 2.

25

Consider the function y = \left(x + 1\right)^{3} + 2\left(x + 1\right)^{3} + 2.

a

Sketch a graph of the function.

b

Hence calculate the exact area bounded by the curve, x-axis, and the lines x = - 2 and x = 1.

26

Consider the function y = \sqrt{x + 1}.

a

Sketch a graph of the function.

b

Hence calculate the exact area bounded by the curve, the x-axis, and the line x = 3.

27

For each of the following functions:

i

Sketch a graph of the function.

ii

Hence determine the exact area bounded by the curve and the x-axis.

a

y = x \left(x - 1\right) \left(x + 3\right)

b

y = \left(x - 1\right) \left(x + 2\right) \left(x + 3\right)

c

y = x \left(x - 1\right)

28

Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.

Find the value of the following:

a

\int_{ - 3 }^{0} f \left( x \right) dx

b

\int_{ - 3 }^{2} f \left( x \right) dx

c

The area enclosed by the curve and the x-axis for x < 0.

d

The area enclosed by the curve and the x-axis.

Area as an absolute value
29

Consider the graph of y = f \left( x \right).

a

Find the value of \int_{0}^{3} f \left( x \right) dx.

b

Find the value of \int_{3}^{8} f \left( x \right) dx.

c

Hence calculate \int_{0}^{8} f \left( x \right) dx.

d

Calculate the area bounded by the function, the x-axis and the y-axis.

e

Write an expression for the area from part (d) as a sum or difference of definite integrals.

f

Write an expression for the area from part (d) using an absolute value sign.

1
2
3
4
5
6
7
8
x
-4
-3
-2
-1
1
2
3
4
y
30

Consider the graph of y = f \left( x \right).

a

Find the value of \int_{0}^{8} f \left( x \right) dx.

b

Calculate the area bounded by the function, the x-axis and the y-axis.

c

Write an expression for the area from part (b) as a sum or difference of definite integrals.

d

Write an expression for the area from part (b) using an absolute value sign.

1
2
3
4
5
6
7
8
9
x
-4
-3
-2
-1
1
2
3
4
y
31

Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.

Find the following:

a

\int_{ - 5 }^{3} f \left( x \right) dx

b

\left|\int_{ - 5 }^{3} f \left( x \right) dx\right|

c

\int_{ - 5 }^{3} \left|f \left( x \right)\right| dx

d

The area enclosed by the curve and the x-axis.

32

Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.

Find the following:

a

\int_{ - 2 }^{7} f \left( x \right) dx

b

\int_{3}^{7} \left( - f \left( x \right) \right) dx

c

\int_{ - 2 }^{7} 2 f \left( x \right) dx

d

\int_{3}^{ - 2 } f \left( x \right) dx + \int_{3}^{7} f \left( x \right) dx

e

\int_{ - 2 }^{7} \left|f \left( x \right)\right| dx

f

The area enclosed by the curve and the x-axis.

33

Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.

Find the following:

a

\int_{ - 4 }^{0} f \left( x \right) dx

b

\left|\int_{ - 4 }^{3} f \left( x \right) dx\right|

c

\int_{ - 1 }^{0} 2 f \left( x \right) dx + \int_{3}^{0} f \left( x \right) dx

d

The area enclosed by the curve and the x-axis.

e

\int_{ - 4 }^{3} \left| f \left( x \right)\right| dx

f

\int_{ - 4 }^{3} \left(f \left( x \right) + x^{2}\right) dx given that the definite integral \int_{ - 4 }^{3} x^{2} dx = \dfrac{91}{3}.

34

Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.

Write the following in terms of A and B:

a

\int_{ - 5 }^{1} f \left( x \right) dx

b

\int_{ - 5 }^{ - 2 } 3 f \left( x \right) dx - \int_{ - 2 }^{1} f \left( x \right) dx

c

\left|\int_{ - 5 }^{1} f \left( x \right) dx\right|

d

\int_{ - 5 }^{1} \left|f \left( x \right)\right| dx

e

\int_{ - 5 }^{1} \left(f \left( x \right) + x\right) dx given that the definite integral \int_{ - 5 }^{1} x \ dx = - 12.

35

Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.

Write the following in terms of A, B and C:

a

\int_{0}^{6} f \left( x \right) dx

b

\int_{6}^{ - 1 } f \left( x \right) dx

c

The area bounded by the curve and the x-axis.

d

\int_{ - 1 }^{6} \left|f \left( x \right)\right| dx

e

\int_{ - 1 }^{6} \left( 2 x - f \left( x \right)\right) dx given that the definite integral \int_{ - 1 }^{6} 2 x \ dx = 35

36

Consider the function f \left( x \right) where x = - 4, 1 and 3 are the only x-intercepts and \int_{ - 4 }^{1} f \left( x \right) dx = 4 and \int_{1}^{3} f \left( x \right) dx = - 7.

Find the following:

a

\int_{ - 4 }^{3} f \left( x \right) dx

b

\int_{ - 4 }^{3} \left|f \left( x \right)\right| dx of f \left( x \right).

c

The area bounded by the curve of f \left( x \right) and the x-axis.

d

\int_{3}^{ - 4 } \left(f \left( x \right) - x^{3}\right) dx, given that \int_{3}^{ - 4 } x^{3} dx = \dfrac{175}{4}.

37

Consider the function f \left( x \right) where x = - 2, 2 and 8 are the only x-intercepts and \int_{ - 2 }^{2} f \left( x \right) dx = - 5 and \int_{ - 2 }^{8} f \left( x \right) dx = 3.

Find the following:

a

\int_{2}^{8} f \left( x \right) dx

b

The area bounded by the curve and the x-axis.

c

\int_{ - 2 }^{8} \left( 2 f \left( x \right) - 6 x\right) \ dx, given that \int_{ - 2 }^{8} x \ dx = 30.

38

Consider the function f \left( x \right) where x= - 6, - 2, 2 and 7 are the only x-intercepts and \int_{ - 6 }^{ - 2 } f \left( x \right) dx = - A , \int_{ - 2 }^{2} f \left( x \right) dx = B and \int_{2}^{7} f \left( x \right) dx = - C.

Write the following in terms of A, B and C:

a

\left|\int_{ - 6 }^{7} f \left( x \right) dx\right|

b

The area bounded by the curve and the x-axis.

c

\int_{ - 2 }^{7} f \left( x \right) dx - \int_{ - 6 }^{2} f \left( x \right) dx

d

\int_{ - 6 }^{2} 2 f \left( x \right) dx + \int_{2}^{7} \dfrac{f \left( x \right)}{2} dx

e

\int_{ - 6 }^{7} \left(3 - f \left( x \right) + x\right) dx, given that \int_{ - 6 }^{7} \left(3 + x\right) dx = \dfrac{91}{2}.

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Outcomes

U34.AoS3.7

informal consideration of the definite integral as a limiting value of a sum involving quantities such as area under a curve and approximation of definite integrals using the trapezium rule

U34.AoS3.9

properties of anti-derivatives and definite integrals

U34.AoS3.14

the concept of approximation to the area under a curve using the trapezium rule, the ideas underlying the fundamental theorem of calculus and the relationship between the definite integral and area

U34.AoS3.20

evaluate approximations to the area under a curve using the trapezium rule, find and verify antiderivatives of specified functions and evaluate definite integrals

U34.AoS3.21

apply definite integrals to the evaluation of the area under a curve and between curves over a specified interval

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