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4.065 Areas under curves

Worksheet
Signed areas
1

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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5
6
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1
2
3
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5
6
y
2

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.

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8
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8
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3

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.

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

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.

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x
-6
-4
-2
2
4
6
y
5

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.

6

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.

7

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.

8

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
9

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

Find the exact area of the shaded region.

10

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

Find the exact area of the shaded region.

11

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

Find the exact area of the shaded region.

-1
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-4
-3
-2
-1
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4
y
12

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

Find the exact area of the shaded region.

13

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

Find the exact area of the shaded region.

14

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

15

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.

16

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.

17

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.

18

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)

19

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
20

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.

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x
-4
-3
-2
-1
1
2
3
4
y
21

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.

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x
-3
-2
-1
1
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y
22

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.

23

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.

24

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}.

25

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.

26

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

27

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}.

28

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.

29

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

3.2.12

interpret the definite integral ∫ {from a to b} f(x)dx as area under the curve y=f(x) if f(x)>0

3.2.13

interpret ∫ {from a to b} f(x)dx as a sum of signed areas

3.2.14

apply the additivity and linearity of definite integrals

3.2.17

develop the formula ∫ {from a to b} f(x)dx= F(b)−F(a) and use it to calculate definite integrals

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