Definite Integrals - area under a curve

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Definite Integrals

Average value of a funcion

equals to the definite integral of f(x) over the interval [a;b] diivided by the upper limit minus the lower limit

Area between two curves

If you have two functions f(x) greater or equal to g(x) over the interval [a;b] the area between them equals to the definite integral of f(x) minus g(x) over the interval [a;b]

Area under a curve

By integration

definite integral of the function y=f(x) over the interval [a;b]

Somtitimes positive sometimes negative

The limits aren't given

you need to find the intersections with the x axis

The function is negative in the interval

you need to put an absolute value at the start or you need to change the sign at the end

The function is not given

you need to find the equation of the function

The solution of the definite integral is the antiderivative of f(x) over the interval [a;b]. Then substitute the upper limit (b) into the integral and subtract the value given by substituting the lower limit (a) into the integral

By approximation

divided area in inner or outer rectangles

Coventions

Changing the upper and the lower limit the integral is the same by putting a minus ahead

when the limits are the same [a;a] the integral equals to 0

Properties of definite integral

if a function is less or equal to another one then the integrals are respectively less or equal to each other like the functions

if theres a constant in a integral it's possible to take it out the integral

the definite integral of the function f(x) plus g(x) equals to the sum of the two integrals of the singular functions in the same order and interval

if a

Math vocabulary

derivative, integral, limit, general symbols, etc.