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iGCSE (2021 Edition)

12.07 Properties of definite integrals

Worksheet
Properties of definite integrals
1

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

2

Consider the function f \left( x \right) where \int_{ - 1 }^{6} f \left( x \right)\, dx = 3. Determine \int_{ - 1 }^{6} \left( 9 f \left( x \right) - 2\right)\, dx.

3

Suppose \int_{4}^{6} f \left( x \right)\, dx = 3. Find the value of:

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

4

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

What property of definite integrals do parts (a) and (b) demonstrate?

5

Consider the even function y = 3 x^{4}.

a

Evaluate the integral \int_{0}^{1} 3 x^{4}\, dx.

b

Hence, find \int_{ - 1 }^{1} 3 x^{4}\, dx.

6

The graph of y = f \left( x \right) has been drawn:

Find the value of a, where a \gt 0, so that \int_{ - a }^{a} f \left( x \right) \, dx = 0.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-3
-2
-1
1
2
3
y
Splitting an interval
7

Suppose \int_{ - 1 }^{2} f \left( x \right)\, dx = 4 and \int_{2}^{8} f \left( x \right)\, dx = 8. Find the value of:

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

8

Suppose \int_{ - 1 }^{3} f \left( x \right)\, dx = 5 and \int_{2}^{3} f \left( x \right)\, dx = 2. Find the value of:

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

9

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

a

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

b

If - 4, 1 and 3 are the only x-intercepts of f \left( x \right), determine \int_{ - 4 }^{3} \left|f \left( x \right)\right|\, dx.

c

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

d

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

10

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

a

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

b

If - 2, 2 and 8 are the only x-intercepts of f \left( x \right), determine the area bounded by the curve f \left( x \right) and the x-axis.

c

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

11

Consider the function f \left( x \right) where \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 expressions for each of the following in terms of A, B and C:

a

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

b

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

c

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

d

\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}.

e

The area bounded by the curve f \left( x \right) and the x-axis if - 6, - 2, 2 and 7 are the only x-intercepts of f \left( x \right).

Applications to area
12

Consider the odd function y = x^{3}.

a

Evaluate the integral \int_{0}^{2} x^{3}\, dx.

b

Hence, write down the answer to \int_{ - 2 }^{2} x^{3}\, dx.

c

Write down the area bounded by the function y = x^{3}, the x-axis and the lines x = - 2 and x = 2.

13

The diagram shows the region bounded by y = \dfrac{1}{x + 3}, x = 0, x = 45 and y = 0:

The region is divided into two parts of equal area, by the line x = k where k > 0.

Find the value of k.

14

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

a

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

b

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

c

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

d

Determine the total area enclosed by the curve and the x-axis.

15

Find the exact area of the shaded region bounded by the line x + y = 3, the x-axis, the y-axis, and the line x=6 shown:

16

Find the exact area of the shaded region bounded by the curve y = 4 - x^{2}, the y-axis, the x-axis and the line x=-3:

17

Find the exact area of the shaded region between the curve y = x \left(x - 1\right) \left(x + 3\right) and the x-axis:

18

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

a

Sketch the graph of the function.

b

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

19

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

a

Sketch the graph of the function.

b

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

20

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

a

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

b

Find the area enclosed by the curve f \left( x \right) and the x-axis.

c

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

d

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

21

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

a

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

b

Find the area enclosed by the curve f \left( x \right) and the x-axis.

c

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

d

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

e

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

f

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

22

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

a

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

b

Find the area enclosed by the curve f \left( x \right) and the x-axis.

c

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

d

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

e

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

23

Consider the function f \left( x \right) shown below. The letters in the shaded regions represent 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 \\ \int_{ - 5 }^{1} x\, dx = - 12

24

Consider the function f \left( x \right) shown below. The letters 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 f \left( x \right) 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 \int_{ - 1 }^{6} 2 x \, dx = 35

25

Consider the function y = f \left( x \right) shown:

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

Find the exact area bounded by the function y=f(x), the x-axis and the y-axis.

2
4
6
8
10
12
14
16
x
-6
-4
-2
2
4
6
y
26

Consider the function y = f \left( x \right) shown:

a

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

b

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

c

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

d

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

-1
1
2
3
4
5
6
7
8
9
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
27

Consider the function y = f \left( x \right) shown:

a

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

b

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

1
2
3
4
5
6
7
8
x
-3
-2
-1
1
2
3
4
y
28

Consider the function y = f \left( x \right) shown:

Which integral has the greatest value?

A
\int_{0}^{8} f(x) \, dx
B
\int_{0}^{1} f(x) \, dx
C
\int_{0}^{2} f(x) \, dx
D
\int_{0}^{5} f(x) \, dx
1
2
3
4
5
6
7
8
9
x
-4
-3
-2
-1
1
2
3
4
y
29

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

The area of region R is 2\text{ units}^2, and the area of region S is 3\text{ units}^2. It is given that \int_{0}^{4} f \left( x \right) \, dx = 10.

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

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Outcomes

0606C14.10

Evaluate definite integrals and apply integration to the evaluation of plane areas.

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