Mathematics in economics
Lecture 10


Lecture 10
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 10
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 10
Definite integral – a use




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Lecture 10
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 10
Definite integral – An area under/above a function




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Graf 22

Lecture 10
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 10
Definite integral – An area under/above a function




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Graf 25

Lecture 10
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 10
Definite integral – An area under/above a function
Problems 3 and 4




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Find:
Solution:
Find:
Solution:

Lecture 10
Problems to solve - 1




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Find:

Lecture 10
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 10
An area between two curves – Problem 1

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



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Graf 26

Lecture 10
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 10
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 10
An area between two curves – Problem 2 – cont.

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

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Graf 27