Lecture8| 1 3.2. Area, Riemann sum, and Definite
Integrals
Consider the area of a region as shown.
We use upper and lower approximations.
Lecture8| 2 We slice into pieces at points
Each has the base length
For each , let (left) (right)
Lecture8| 3
Upper Approximation (highest boxes)
We replace each with the rectangle of height
The approximated area is
Lower Approximation (lowest boxes)
We replace each with the rectangle of height
The approximated area is
Lecture8| 4 If the region is sliced into more pieces:
The area should be !
Lecture8| 5 Area of a Region. Let be continuous on . Assume .
The area of the region as shown is
base length, height
Lecture8| 6 EXAMPLE. Find the area
, for
Lecture8| 7 Other Quantities
Works
for a constant force moving an object by distance . Here, either or .
For variable force, we approximate
Lecture8| 8
Charges
for a constant current over a time interval . Again, we can have or .
For variable current, we approximate
These sums give the precise values of and by taking , , respectively.
Lecture8| 9 Definition. Let be a function on . We do not require continuous nor .
We partition by points
Let base length, a point in
each base .
The Riemann sum of for this partition is
If the following limit exists
is called integrable and the limit is denoted by
Lecture8| 10 is called the definite integral of from to .
If and is continuous, of course
If variable force, then
If current, then
Lecture8| 11 Theorem. If is continuous on , then is
integrable on . That is exists.
For area, we now have the following result.
Area as a definite integral.
If is continuous on and , the area of region bounded by the graph , -axis, lines and is
Lecture8| 12 EXAMPLE. Express each of the following areas as a definite integral.
Lecture8| 13
Lecture8| 14 EXAMPLE. Evaluate the definite integral
,
Lecture8| 15
Note the value of integral is negative, so it does not represent area.