1. 5.3 Definite Integrals and Antiderivatives Greg Kelly, Hanford High School, Richland, Washington

2. Page 269 gives rules for working with integrals, the most important of which are: 1. If the upper and lower limits are equal, then the integral is zero. 2. Reversing the limits changes the sign. 3. Constant multiples can be moved outside.

3. 1. If the upper and lower limits are equal, then the integral is zero. 2. Reversing the limits changes the sign. 3. Constant multiples can be moved outside. 5-6. Integrals can be added and subtracted.

4. 5-6. Integrals can be added and subtracted. 10. Intervals can be added (or subtracted.)

5. The average value of a function is the value that would give the same area if the function was a constant:

6. The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal the average value. Average value of a function If f is continuous on then at some point c in, 

7. The math professor's six-year-old son knocks at the door of his father's study. "Daddy", he says. "I need help with a math problem I couldn't do at school." "Sure", the father says and smiles. "Just tell me what's bothering you." "Well, it's a really hard problem: There are four ducks swimming in a pond, when two more ducks come and join them. How many ducks are now swimming in the pond?" The professor stares at his son with disbelief: "You couldn't do that?! All you need to know is that 4 + 2 = 6!" "Do you think, I'm stupid?! Of course, I know that 4 + 2 = 6. But what does this have to do with ducks!?"