Areas of Domains

1
Areas of Domains
Areas of Domains between the Graph of a Function
and the x-Axis Areas of Domains between the
Graphs of two Functions Integrating with respect
to y
2
Area of the Domain under a Graph
The picture on the right shows a middle rule
approximation of the domain bounded by a positive
function f. The corresponding Riemann sum
gives the combined area of the green boxes in the
picture.
Example
The area of the yellow
domain under the graph of the sine curve and
above the x-interval 0,p is
3
Functions taking positive and negative values
The domains between the x-axis and the graphs of
the functions f(x) and f(x) have the same
area.
If the function f takes also negative values,
then the area of the domain between the graph of
f and the x-axis is given by the integral
Example
The area between the sin curve and the x-axis
interval 0,2p is
4
Area of a Domain between Graphs
The area of the domain between the graphs of two
functions f(x) and g(x) for a x b is
given by the integral
Example
The area of the domain
between the graphs of the sine and the cosine
curves for p/4 x 5p/4 is
Because sin(x) cos(x) when p/4 x 5p/4.
5
Integrating with Respect to y
Example
It is a technical simplification to view these
functions as functions of y rather than x.
Integrating with respect to y instead of x
allowed us to express the area as a single
integral instead of a sum of two integrals.
The functions are then x y 2 and x y2.
The domain in question is now the domain between
the graphs of these functions of y for 0 y
2. The area is hence