Definite Integrals

derivative, integral, limit, general symbols, etc.

if a<c<b then the definite integral over the interval (a;b) is equal to the definite integral over the interval (a;c) plus the definite integral over the interval (c;b)

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

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

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

By approximation

divided area in inner or outer rectangles

By integration

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

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

The function is not given

you need to find the equation of the function

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 limits aren't given

you need to find the intersections with the x axis

Somtitimes positive sometimes negative

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]

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