Calculation of areas – Definite Integral

Calculation of areas – Definite Integral

Calculation of areas – Definite Integral

 

Definite Integral of a function is used to calculate the areas of flat regions, which are delimited by curves and straight lines.

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Definite Integral

Bioprofe | Calculation of areas – Definite Integral | 01

 

Areas of flat regions:

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Definite Integral

Bioprofe | Calculation of areas – Definite Integral | 02

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Definite Integral

Bioprofe | Calculation of areas – Definite Integral | 03

 

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Definite Integral

Bioprofe | Calculation of areas – Definite Integral | 04

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integral | 05

 

 

Areas limited by several functions:

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integral | 06

 

This area is calculated as follows:

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 07

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 08

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 09

 
 

Functions whose graphs are waveform:

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 10

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 11

 

Examples:

 

Area delimited by the parabola y=x² and the line y=1

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 12

 

Integration interval:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 13

 

Therefore:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 14

 
 

Area delimited by the parabola y=x², the line y=-x+2 and the ox axis

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 15

 

 

Points of intersection:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 16

 

It is applied the following formula:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 17

 

Integration interval:
x=1
x=-2

 

Therefore:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 18

 

 

Area delimited by the parabola y=-x², the lines x=-1, x=2 and the ox axis

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 19

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 20

 

 

Area delimited by the function y=x³, the lines x=-1, x=2 and the ox axis

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 21

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 22

 

It is solved:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 23

 
 

Area delimited by the functions y=e^x, y=e^-x , the lines x=-1, x=1 and the ox axis

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 24

 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Infinitesimal calculus | Calculation of areas

Bioprofe | Calculation of areas – Definite Integrals | 25

 

 

 

 

You can download the App BioProfe READER to practice this theory with self-corrected exercises.

 

 

Books that we recommend to extend knowledge of the subject

 

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