There are two main methods to calculate the area underneath and between curves: the Riemann sum and using integrals. The Riemann sum approximates the area by breaking it up into smaller rectangles and adding their areas. Using integrals to calculate the area involves finding the intersection points to determine the interval, identifying the upper and lower functions, and taking the integral of the area between the functions.

1. HOW TO CALCULATE THE AREA
UNDERNEATH AND BETWEEN CURVES
C A I T I E B R O W N
A Glimpse Into Integration

2. What is Integration?
 Anti-differentiation (the
opposite of taking the
derivative)
 An integral is a function of
which a given function is the
derivative (yields that function
when differentiated, and which
may express the area under the
curve of a graph of the
function.)

3. Methods
There are 2 main methods used to calculate the area
under/between curves:
• The Riemann Sum
OR
• Using integrals

4. The Riemann Sum
 Hard to find the area underneath
the curve alone, so break it up into
smaller rectangles which you can
easily find the area of, then add
them up.
 Most accurate answer will come
from using small rectangles with
the midpoints of their base
intersecting the curve
 The width of each rectangle is 1/2.
and the height will be the value of
the function at the midpoint of the
rectangle. In this case the area of
the region will be approximated by
this equation:
The shaded region
is between the
function f(x)=x2 +1
and the x-axis and
on the interval
[0,2]
Break the area up
into rectangles

5. Integrals
To
determine
the area
bound by
these two
curves:
•Find the points where the lines
intersect (to find the interval) by
making them equal each other.
2x2+10=4x+16
2x2-4x-6=0
2(x+1)(x-3)=0
In this case it is [-1,3]
•The image of the graph allows
identification of the upper/lower
function
•Then, sub functions into the equation,
expand, and take the integrals