1. 5.1 Approximating and Computing Area Fri Jan 15
Evaluate each summation 1) 2)

2. Distance and velocity Distance traveled = velocity x time elapsed
If velocity changes (is a function):
Then distance traveled = f(t) x dt
or the area under the graph of f(t) over [t1, t2]
Note: in this scenario we are going from a rate to not a rate

3. Area under a curve
So finding the area under a curve allows us to calculate how much has accumulated over time (common AP free response)
First we will discuss ways to approximate this accumulation

4. Approximating Area
Although the area under a graph is usually curved, we can use rectangles to approximate the area.
More rectangles = better approximation
Draw graph here

5. 3 types of rectangle approx.
The 3 approximations only differ in where the height of each rectangle is determined
Left-endpoint: use the left endpoint of each rectangle to determine height
Right-endpoint: use the right endpoint of each rectangle
Midpoint: use the midpoint of the endpoints

6. Rectangle Approx. 1) Determine and each interval
2) Use the correct endpoint to find the height of each rectangle
3) Add all the heights together
4) Multiply sum by the width ( )

7. endpoint formulas
The formula for the Nth right-endpoint approximation:
The formula for the Nth left-endpoint approx:

8. Ex
Calculate for on the interval [1,3]

9. Ex
Calculate for the same function [1,3]

10. Ex
Calculate for on [2,4]

11. Interpretations Which approximation is the best?
Depends on the function
Left and Right endpoints can overestimate or underestimate depending on if the function increases or decreases
Midpoint is typically the safest because it does both so it averages out

12. Summation Review
Summation notation is standard for writing sums in compact form

13. Approximations as Sums

14. Summation Theorems
If n is any positive integer and c is any constant, then:
Sum of constants
Sum of n integers
Sum of n squares
These theorems allow us to use a formula when evaluating sigma notation. Simply plug in the n value into the formula to evaluate these sums. i^3 is in your book

15. Summation Theorems For any constants c and d,
This theorem states that we can pull out any constants in front of the sigma notation. We can also separate a sigma notation into two separate sums.

16. Computing Actual Area
So far all we’ve been doing is adding areas of rectangles, where does the calculus come in?
As the # of rectangles approach infinity, we can find the actual area under a curve

17. Closure
Approximate the area under f(x) = x^2 under the interval [2, 4] using left-endpoint approximation and 4 rectangles
HW: p.296 #1-27 odds

18. 5.1 Computing and Approximating Area Tues Jan 19
Do Now
Approximate the area under f(x) = -x^2 + 4 on the interval [-1, 1] using 4 right endpoint rectangles

19. HW Review

20. Worksheet
If time

21. Closure
How can we approximate the area under a graph? How could we compute the exact area? How would this be useful?
HW: none

22. Theorem – not covered on AP
If f(x) is continuous on [a, b], then the endpoint and midpoint approximations approach one and the same limit as
There is a value L such that

23. Ex 1
Find the area under the graph of f(x) = x over [0,4] using the limit of right-endpoint approx.

24. Ex 2
Let A be the area under the graph of f(x) = 2x^2 – x + 3 over [2,4]. Compute A as the limit