finding area using definite integral
the area between 2 functions 
in an interval 
in which 
or 
is defined as:
in an interval in which or is defined as:
notice that the definition says in which or which means that if the functions cross each other in that interval and the lower function becomes the upper and the upper function becomes the lower then the area cant be calculated using this single formula but we split the interval into multiple and add the results
in which or which means that if the functions cross each other in that interval and the lower function becomes the upper and the upper function becomes the lower then the area cant be calculated using this single formula but we split the interval into multiple and add the resultsconsider the area 
in the interval 
between 
and the 
axis (
)
we calculate using the formula, i didnt mind the fact that the result should be turned to its absolute value because i know the
in the interval between and the axis ()
we calculate using the formula, i didnt mind the fact that the result should be turned to its absolute value because i know the
consider the area 
in red between the polynomials
we split the area into 2 because if we tried to find the area without splitting we would be adding a negative value to a positive value whereas the areas should be positive, we wouldnt mind this when working with functions that dont cross because we can always take the absolute value of the result
so we find
and 
and the total area would be 
, assume 
, would be the area from 
and the area from
for we take 
because
for we take because
therefore
in red between the polynomials
we split the area into 2 because if we tried to find the area without splitting we would be adding a negative value to a positive value whereas the areas should be positive, we wouldnt mind this when working with functions that dont cross because we can always take the absolute value of the result
so we find
and and the total area would be , assume , would be the area from and the area from
for
we take because
for
we take because
therefore
notice that these examples provide you with a plot of the functions so its easier to find the area, in general you wouldnt have the plot and you have to find the intersection points yourself so that you can integrate and find the area, see the following exercise for example
find the area between 
first we find the intersection points so that we know how to split the integration
we found that we have intersections at
, so we find the integrals in the intervals 
area is
first we find the intersection points so that we know how to split the integration
we found that we have intersections at
, so we find the integrals in the intervals
area is