Properties of Integrals

OVERVIEW:

1. Developing Expert Voices (DEV) Project
2. StoryTools
3. The Fundamental Theorem of Calculus
4. Properties of Integrals - Introduction
5. Constant Multiple Rule
6. Sum and Difference Rule
7. Additive Interval Rule
8. Inequality Rule
9. End Notes

Developing Expert Voices (DEV) Project and StoryTools (Slide 1)

Timelines and sample questions were due today.

Continuing our discussion about our DEV projects, Mr.K provided us a link, http://cogdogroo.wikispaces.com/Dominoe+50+Ways, which is a list of web tools you can use for any project or assignment.


The Fundamental Theorem of Calculus (Slides 2 and 3)

Slides 2 and 3 is a summary of what we did in the previous class. Check the previous scribe post for an explanation.


Properties of Integrals (Slides 4)


Slide 4 summarizes the properties of integrals. (It's the most important slide in this scribe post, in my opinion.) Rules were given; then explained.

Constant Multiple Rule (Slide 5)

A function with a constant inside it can be factored out. Knowing this...

Integrating a function with a constant not factored out = integrating a function with a constant factored out.

For example, doubling the sum of areas equals the sum of all the areas doubled, if that makes better sense.


Sum and Difference Rule (Slide 6)

The integral of a sum is the sum of its integrals. In other words, when you are asked to integrate a polynomial function, you can integrate the terms inside the polynomial function, as seen in slide 6. However, the answer isn't complete in slide 6. Answer is 94/3.

Additive Interval Rule (Slide 7)

The area under a function of a graph in a given interval can be seen as the sum of smaller areas.

In this slide, using the Additive Interval Rule, the red area (an integral) plus the blue area (another integral) gives the green (total) area which is the integral for the function in the interval [-2, 2]. Remember that an integral is the area underneath the function of a graph.

Inequality Rule (Slide 9)
If a function is less than or equal to another function in the same interval, then the area underneath the former function is less than or equal to the area underneath the latter function.


End Notes:

• 6.1 The Definite Integral Again odd
• Post your DEV timeline on the blog by Sunday
• Next scribe is benofschool