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Standard level

11.04 Properties of definite integrals

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

Area under a curve
2

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.

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2
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5
6
x
1
2
3
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5
6
y
3

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
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8
x
1
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5
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8
y
4

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.

5

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

Find the exact area of the shaded region.

6

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

Find the exact area of the shaded region.

7

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

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