Mathematics in economics
Lecture 7
Mgr. Jiří Mazurek, Ph.D.
Mathematics in Economics/PMAT

Lecture 7
Definite integral




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Newton´s definite integral:
In the definition above, F is a primitive function to f,
and a and b are the limits of the integral.
The result of definite integral is not a function, but a number!

Lecture 7
Definite integral – elementary properties




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Generally, when computing definite integral, we use the
same table of elementary integrals as for an indefinite
integral.
The elementary properties of the definite integral:

Lecture 7
Definite integral – a use




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The definite integral can be used to calculation of:
• The square under or above given function,
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• The length of a curve,
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• The volume of a 3D object,
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• The area of a 3D object.

Lecture 7
Definite integral – An area under/above a function




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Find:
Solution:
What does the number 7/3 mean?
It is the area below the function f(x) on the interval (1,2),
See the next slide for a picture.

Lecture 7
Definite integral – An area under/above a function




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Graf 22

Lecture 7
Definite integral – An area under/above a function




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Find an area bounded by functions:               , axis x,
x = -1 and x = 2.
Solution: We must divide the interval of integration
(-1,2) into two intervals: (-1,0) and (0,2) (WHY?):

Lecture 7
Definite integral – An area under/above a function




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Graf 25

Lecture 7
Definite integral – An area under/above a function
Problem 2




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Find:
Solution:
This result means that the area under the function on
 the interval (0,3) is 9.
Important note: if a function is positive on the interval
of integration, then the result will be a positive number.
However, for a negative function the result will be
negative!

Lecture 7
Definite integral – An area under/above a function
Problems 3 and 4




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Find:
Solution:
Find:
Solution:

Lecture 7
Definite integral – per partes




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Per partes method:
Example:

Lecture 7
Definite integral – substitution




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A substitution in an definite integral, example:

Lecture 7
Problems to solve - 1




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Find:

Lecture 7
An area between two curves




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Let f(x) and h(x) be two curves, S an area between them
And a and b their intersections.
Then S is given as follows:

Lecture 7
An area between two curves – Problem 1

Find an are between two curves:              and             .
A picture:



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Graf 26

Lecture 7
An area between two curves – Problem 1 cont.

Find an are between two curves:              and             .
Solution:
First, we find intersections:                , hence x = 0 and x = 1.
 Now, we can use the integral formula for the area:




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Lecture 7
An area between two curves – Problem 2

Find an are between two curves:              and             .
Solution:
First, we find intersections:                , hence x = 0 and x = 2.
 Now, we can use the integral formula for the area:




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Lecture 7
An area between two curves – Problem 2 – cont.

Find an are between two curves:              and             .
A graph:

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Graf 27

Lecture 7
A volume of a solid of revolution


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We assume that a solid is generated by rotating a plane
curve around x axis.
In such a case, the volume of a solid is given as:

Lecture 7
A volume of a solid of revolution – Problem 1


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Find a volume of a solid generated by a curve               ,
rotating around x axis on the interval  (0,3).
Solution:
Note: the solid is called a rotational parabolloid.

Lecture 7
A volume of a solid of revolution – Problem 2


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Find a volume of a solid generated by a curve               ,
rotating around x axis on the interval  (1,2).
Solution:
Note: the solid is also a rotational parabolloid.

Lecture 7
A volume of a solid of revolution – Problem 3


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Find a volume of a solid generated by a curve               ,
rotating around x axis on the interval  (1,4).
Solution:
Note: the solid is called a rotational hyperbolloid.

Lecture 7
Problems to solve - 1


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Find a volume of a solid generated by a curve               ,
rotating around x axis on the interval  (1,2).
Find an are under/above the curve           on the interval (1,3)
Find an are under/above the curve:
Find an are under/above the curve:

Lecture 7
Problems to solve - 2


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Find an are under/above the curve:
Find an are under/above the curve:
Find an are under/above the curve:
Find an are under/above the curve:

Lecture 7
Problems to solve - 3


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Find an are between two curves:
Find an are between two curves:
Find an are between two curves:
Find a volume of a rotational solid:                on interval (0,1).

Lecture 7
Thank you for your attention


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