Topics
7. Introduction to Integral Calculus
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AustraliaNSW
Year 12

7.05 Definite integrals and the area under a curve

Lesson
Worksheet
Practice
Worksheet
Geometric solutions to definite integrals
1

Find the exact value of the following definite integrals using the graphs provided:

a

\int_{1}^{8} \left(10 - x\right) \,\, dx

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b

\int_{3}^{5} \left( 2 x + 1\right) \, dx

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c

\int_{ - 6 }^{6} \sqrt{36 - x^{2}} \, dx

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2

For each of the following graphs, find the exact value of \int_{0}^{16} f \left( x \right) dx:

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3

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

a

Find the value of:

i
\int_{0}^{4} f \left( x \right)\, dx
ii
\int_{4}^{6} f \left( x \right)\, dx
iii
\int_{6}^{8} f \left( x \right)\, dx
b

Hence, calculate the area bounded by the function and the x-axis.

-1
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Definite integrals
4

Evaluate the following:

a
\int_{0}^{3} \left( 4 x + 5\right)\, dx
b
\int_{ - 2 }^{0} \left( 10 x + 4\right)\, dx
c
\int_{ - 4 }^{5} \left( - 8 x + 3\right)\, dx
d
\int_{ - 10 }^{ - 6 } \left( - 5 x + 8\right)\, dx
e
\int_{ - 2 }^{0} 9 x^{2}\, dx
f
\int_{ - 1 }^{3} 9 x^{2}\, dx
g
\int_{0}^{4} 5 x^{\frac{3}{2}}\, dx
h
\int_{3}^{4} \left( 2 x + 3\right)^{3}\, 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_{6}^{11} \sqrt{x - 2}\, dx
m
\int_{3}^{6} \left(\sqrt{x - 2} + 5\right)\, dx
n
\int_{ - 3 }^{6} x \left(x + 3\right) \left(x - 6\right)\, dx
o
\int_{ - 4 }^{2} x \left(x - 4\right)\, dx
p
\int_{ - 4 }^{2} \left(\left(x + 2\right)^{3} + 3\right)\, dx
5

Evaluate the following:

a
\int_{2}^{3}\dfrac{2x^{2}-4x}{6x}\,dx
b
\int_{2}^{4}\dfrac{3x^{4}+2x^{2}}{x^{2}}\,dx
c
\int_{-4}^{-1}\dfrac{-2x^{5}+x^{4}}{3x^{2}}\,dx
d
\int_{1}^{2} \dfrac{x^{5} - x^{ - 2 }}{x^{2}}\, dx
Areas
6

Find the exact area of the shaded region under the curve y = 6 x^{2}.

7

Find the exact area of the shaded region under the curve y = x^{2} + 6.

8

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

a

Sketch the graph of the function.

b

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

9

Consider the function y = \dfrac{1}{x - 2} + 3.

a

Sketch the graph of the function.

b

Hence, determine the exact area bounded by the curve, the x-axis, and the lines x = 4 and x = 6.

10

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

a

Sketch the graph of the function.

b

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

11

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

a

Sketch the graph of the function.

b

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

12

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

a

Sketch the graph of the function.

b

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

13

Consider the function y = 2 x + 3.

a

Sketch the graph of the function.

b

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

14

Consider the function y = - 2 x + 8.

a

Sketch the graph of the function.

b

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

15
a

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

b

Hence, determine the area bounded by the curve y = \left(x - 3\right) \left(x - 9\right), the x-axis and the bounds x = 3 and x = 9.

16
a

Calculate \int_{ - 4 }^{3} 5\, dx.

b

Hence, determine the area bounded by the curve y = 5, the x-axis and the lines x = - 4 and x = 3.

17
a

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

b

Hence, determine the area bounded by the line y = 2 x - 8, the x-axis and the lines \\ x = - 2 and x = 4.

18
a

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

b

Hence, determine the area bounded by the curve y = - \left(x + 2\right) \left(x + 8\right), the x-axis and the lines x = - 8 and x = - 2.

Fundamental theorem of calculus
19

Evaluate the following:

a
\dfrac{d}{d x} \int_{2}^{x} \left( 6 t^{2} - 8 t + 3\right)\, dt
b
\dfrac{d}{d x} \int_{ - 2 }^{x} \left(\sqrt{t + 3} - 6 t\right)\, dt
c
\dfrac{d}{d x} \int_{0}^{x} \left(t^{\frac{2}{3}} - t^{2}\right)\, dt
d
\dfrac{d}{d x} \int_{10}^{x} \left( - 4 t^{3} + 4 t - 7\right)\, dt
e
\dfrac{d}{d x} \int_{ - 6 }^{x} \dfrac{2^{t} t^{5}}{3}\, dt
f
\dfrac{d}{d x} \int_{13}^{x} \left(\dfrac{3}{t^{2}} - \dfrac{4}{t^{3}}\right)\, dt
20

Consider the function f \left( t \right) = t^{4} + 8 t^{2} + 20. Find \int_{1}^{x} f \left( t \right)\, dt.

21

The function f has an antiderivative F, and F \left( 3 \right) = 4.

a

Express \int_{3}^{x} f \left( t \right)\, dt in terms of F and x.

b

Find \dfrac{d}{d x} \int_{3}^{x} f \left( t \right) dt.

22

Consider the function f \left( t \right) = - 4 t.

a

Find \int_{6}^{x} f \left( t \right)\, dt.

b

Hence, find \dfrac{d}{d x} \int_{6}^{x} f \left( t \right)\, dt.

c

What can we conclude about \dfrac{d}{d x} \int_{6}^{x} f \left( t \right)\, dt?

23

Consider the function f \left( t \right) = 12 t + 9.

a

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

b

Hence, find \dfrac{d}{d x} \int_{ - 3 }^{x} f \left( t \right)\, dt.

24

Consider the expression \dfrac{d}{d x} \int_{k}^{x} \dfrac{1}{\sqrt{t}}\, dt.

a

What restrictions must be on the value of k for the integration to be possible?

b

Find \int_{k}^{x} \dfrac{1}{\sqrt{t}}\, dt.

c

Hence, find \dfrac{d}{d x} \int_{k}^{x} \dfrac{1}{\sqrt{t}}\, dt.

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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-7

applies the concepts and techniques of indefinite and definite integrals in the solution of problems

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