1. AP Calc: Chapter 5 The beginning of integral fun…
Section 5.1 – Warm Up
Complete the Quick Review on page 1 of your packets 

2. Section 5.1 - Estimating with Finite Sums
Objective: Estimating distance traveled, using the Rectangular Approximation Method (RAM), and Finding the volume of a sphere.

3. Distance Traveled
Explore the problem: A train moves along a track at a constant rate of 75 miles per hour from 7:00 am to 9:00 am. What is the total distance traveled by the train?
Recall: distance = rate * time
Graph: What shape is the graph? How do you find the area of such a shape?

4. Rectangular Approximation Method (RAM)
NOTE: RAM is the same thing as finding Riemann Sums like we did last year (and our Michigan Maps!).
We can distinguish between the 3 types (left endpoint, right endpoint, and midpoint) by the following abbreviations: LRAM (left), MRAM (midpoint), RRAM (right).

7. Example 1:
A particle starts at x = 0 and moves along the x-axis with velocity v(t) = t2 for time . Where is the particle at t = 3?
Methods to approximate?

8. Example 2:
The graph of y = x2sin x on the interval [0, 3] is crazy. Lets Graph it and then estimate the area under the curve from x = 0 to x = 3.

9. Generalizations For an increasing function LRAM MRAM RRAM
For a decreasing function

10. 5.5 trapezoidal rule

15. Section 5.2 Definite Integrals

16. Anatomy of an Integral
Function
Stop
Integral
Change in x
Start

17. Example 1:
Express the area of the shaded region below with an integral.

18. Definite Integrals and Area
Example 2:
Evaluate by drawing a picture

19. :
Example 3 .
Find the exact area of

20. Example 4
Evaluate the integral

21. .
Ex. 5:
Use area to find

22. For any integrable function
= (area above the x-axis) – (area below the x-axis)

23. Example 6: Evaluate the following integrals without a calculator given that

24. Integrals on a calculator!!!
Evaluate the following integrals numerically on your calculators: a) b) c)
Fn int (function,x,upper,lower)

25. Section , 5.5 Refresher