1) The document introduces the concept of definite integrals and their relationship to antiderivatives. 2) It states that the definite integral from a to b is equal to the area under the curve between those bounds, and discusses some basic rules for manipulating definite integrals. 3) The Mean Value Theorem is presented as stating that the average value of a function on an interval must be equal to the function's value at some point within that interval. 4) It is noted that differentiation and integration are inverse operations, and that the derivative of an integral is the original function, up to a constant term.

1. Introduction to Definite Integrals and Antiderivatives Unit 6 Lesson 3 E. Alexander Burt Potomac School

2. Background: Recall the last lesson: The definite integral from a to b is equal to the area between f(x) and the x axis bound by x=a on one side and x=b on the other http://www.teacherschoice.com.au/maths_library/calculus/area_under_a_curve.htm

3. Extending the idea: Definite Integral Rules Reversing limits is negative integral

4. Same Limits no area

5. Constant Multiple

6. Sum and Difference

7. Contiguous Areas Add

8. Extending the Idea 2: Mean Value Theorem The mean (or average) value of a continuous function between a and b is Note that this is just the definite integral divided by the “base”

9. The function will equal its own average value at at least one point in the interval [a,b]

10. The derivative and the integral are opposite operations In the same sense that multiplication and division are opposite, with the right limits, derivatives and definite integrals are also opposites Note the variable change – this is to avoid confusion with too many x variables. The idea remains: the derivative of the integral is the function!

11. The definite integral is an antiderivatvie Since they are opposite operations, the definite integral must be related to the antiderivative. It is: Same variable change as last time.

12. Note the +c because the addition of a constant does not change the derivative.

13. Using Antiderivatives To Evaluate Definite Integrals Reversing the process from the last slide, we can now show that: Or, more generally: