1. Georg Friedrich Bernhard Riemann
Chapter 5 – Integrals
5.2 The Definite Integral
5.2 The Definite Integral
Georg Friedrich Bernhard Riemann

2. Review - Riemann Sum
When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. The width of a rectangle is called a subinterval. The entire interval is called the partition.
subinterval
partition
Subintervals do not all have to be the same size.
5.2 The Definite Integral

3. Example 1 – pg. 382 # 4
If you are given the information above, evaluate the Riemann sum with n=6, taking sample points to be right endpoints. What does the Riemann sum illustrate? Illustrate with a diagram.
Repeat part a with midpoints as sample points.
5.2 The Definite Integral

4. Idea of the Definite Integral
is called the definite integral of
over
If we use subintervals of equal length, then the length of a subinterval is:
The definite integral is then given by:
5.2 The Definite Integral

5. Definite Integral in Leibnitz Notation
Leibnitz introduced a simpler notation for the definite integral:
Note that the very small change in x becomes dx.
5.2 The Definite Integral

6. Explanation of the Notation
upper limit of integration
Integration
Symbol
integrand
variable of integration
(dummy variable)
lower limit of integration
It is called a dummy variable because the answer does not depend on the variable chosen.
5.2 The Definite Integral

7. Theorem (3)
If f is continuous on [a, b], or if f has only a finite number of jump discontinuities, then f is integrable on [a, b]; that is, the definite integral exists.
5.2 The Definite Integral

8. Theorem (4)
Putting all of the ideas together, if f is differentiable on [a, b], then where
5.2 The Definite Integral

9. Example 2
Use the midpoint rule with the given value of n to approximate the integral. Round your answers to four decimal places.
5.2 The Definite Integral

10. Evaluating Integrals using Sums
5.2 The Definite Integral

11. Example 4 – Page 377 #23
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
5.2 The Definite Integral

12. Example 5
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
5.2 The Definite Integral

13. Example 6 – Page 383 # 29
Express the integral as a limit of Riemann sums. Do not evaluate the limit.
5.2 The Definite Integral

14. Example 7 – page 383 # 17
Express the limit as a definite integral on the given interval.
5.2 The Definite Integral

15. Example 8 – page 385 # 71 Express the limit as a definite integral.
5.2 The Definite Integral

16. Properties of the Integral
5.2 The Definite Integral

17. Properties Continued
5.2 The Definite Integral

18. Example 9 – page 384 # 62
Use Property 8 to estimate the value of the integral.
5.2 The Definite Integral

19. Example 10 – page 384 # 37
Evaluate the integral by interpreting it in terms of areas.
5.2 The Definite Integral

20. Example 11 – page 383 # 28 Work in groups to prove the following:
5.2 The Definite Integral

21. What to expect next…
We will be evaluating Leibnitz integrals using the idea of antiderivatives and the fundamental theorem of calculus.
5.2 The Definite Integral