1. Sr. lecturer in mathematics KHEMUNDI COLLEGE , DIGAPAHANDI.
Riemann Sums
and
the Definite Integral
Prepared By
Sri. Santosh Kumar Rath.
Sr. lecturer in mathematics
KHEMUNDI COLLEGE , DIGAPAHANDI.

2. Review We partition the interval into n sub-intervals
Evaluate f(x) at right endpoints of kth sub-interval for k = 1, 2, 3, … n
f(x)

3. Review a b
Sum
We expect Sn to improve thus we define A, the area under the curve, to equal the above limit.
f(x)

4. Riemann Sum
Partition the interval [a,b] into n subintervals a = x0 < x1 … < xn-1< xn = b
Call this partition P
The kth subinterval is xk = xk-1 – xk
Largest xk is called the norm, called ||P||
Choose an arbitrary value from each subinterval, call it

5. Riemann Sum Form the sum This is the Riemann sum associated with
the function f
the given partition P
the chosen subinterval representatives
We will express a variety of quantities in terms of the Riemann sum

6. The Riemann Sum Calculated
Consider the function 2x2 – 7x + 5
Use x = 0.1
Let the = left edge of each subinterval
Note the sum

7. The Riemann Sum We have summed a series of boxes
If the x were smaller, we would have gotten a better approximation
f(x) = 2x2 – 7x + 5

8. The Definite Integral
The definite integral is the limit of the Riemann sum
We say that f is integrable when
the number I can be approximated as accurate as needed by making ||P|| sufficiently small
f must exist on [a,b] and the Riemann sum must exist

9. Example
Try
Use summation on calculator.

10. Example
Note increased accuracy with smaller x

11. Limit of the Riemann Sum
The definite integral is the limit of the Riemann sum.

12. Properties of Definite Integral
Integral of a sum = sum of integrals
Factor out a constant
Dominance

13. Properties of Definite Integral
Subdivision rule
f(x) a c b

14. Area As An Integral The area under the curve on the interval [a,b] A
f(x) A a c

15. Distance As An Integral
Given that v(t) = the velocity function with respect to time:
Then Distance traveled can be determined by a definite integral
Think of a summation for many small time slices of distance