1. Riemann Sums and the Definite Integral
Lesson 5.3

2. Why? Why is the area of the yellow rectangle at the end = b a
The yellow rectangle is the sum of the differences between right and left sums! a b

3. 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)

4. Limit of a Sum
a b
For a function f(x), the area under the curve from a to b is where x = (b – a)/n and
Consider the region bounded by f(x) = x2 the axes, and the lines x = 2 and x = 3

5. Limit of a Sum
Now So

6. Limit of a Sum
Continuing …

7. 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)
Look at Java demo

8. 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

9. 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

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

11. 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

12. 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

13. Example
Try
We will use 4 "slices" n = 4
Use summation on calculator.

14. Example
Note increased accuracy with smaller x

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

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

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

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

19. 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

20. Assignment
Section 5.3
Page 314
Problems: 3 – 49 odd