1. 27. Sections 5.1/7.1 Approximating and Computing Area
Table of Contents
27. Sections 5.1/7.1 Approximating and Computing Area

2. Approximating and Computing Area
Essential Question – What is the other major topic of calculus?

3. 3 main concepts
Limits
Derivatives
Integrals

4. Integrals
Describe how the instantaneous changes of functions (derivatives) can accumulate over an interval to produce the function

5. Finding area Break region into subintervals (strips)
These strips resemble rectangles
Sum of all the areas of these “rectangles” will give the total area

6. Example
A particle starts at x=0 and moves along the x-axis with velocity v(t)=t2. Where is the particle at t=3?
Each interval is ¼ width. Find height at each midpoint of interval. Multiply height times width to get area. Sum all the areas.
Add all 1/256+9/256+25/256+49/256+81/ / / / / / / /256 = 8.98
Subinterval
[0, ¼]
[¼, ½]
[½,3/4]
[3/4, 1]
etc
Midpoint
1/8
3/8
5/8
7/8
Height (t2)
1/64
9/64
25/64
49/64
Area
1/256
9/256
25/256
49/256

7. Rectangular Approximation Method (RAM)
In the last example, we used the midpoint RAM (MRAM) (because we used the midpoint of the interval)
We can also use the left hand endpoint (LRAM)
Or the right hand endpoint (RRAM)
If the function is monotonic (either increasing or decreasing), the actual area lies somewhere between LRAM and RRAM
The more rectangles you make, the better the approximation

8. Rectangular Approximation Method (RAM)
If a function is increasing, LRAM will underestimate the area and RRAM will overestimate it.
If a function is decreasing, LRAM will overestimate the area and RRAM will underestimate it

9. Trapezoid Approximation
Another approximation we can use (and probably the best) is trapezoids.
Trapezoids give an answer between the LRAM and RRAM
The formula for the area of a trapezoid is ½(y1+y2)(x)
The y values in the middle will be doubled and the ones on the end will be one time

10. Example Find the area under y=x2 from x=0 to x=3, use width of ½ LRAM
MRAM
RRAM
Trap

11. Example Find the area under y=x2+2x-3 from x=0 to x=2, use width of ½
LRAM
MRAM
RRAM
Trap

12. Summation Notation
means sum
K tells where to start and end summing

13. How many rectangles should we make?
The estimate of area gets more and more accurate as the number of rectangles (n) gets larger
If we take the limit as n approaches infinity, we should get the exact area
We will take more about this tomorrow…..

14. Book notation If interval is 3 units long and you have 6 subintervals,
Each subinterval will be 3/6 or ½ wide.

15. Assignment
Pg. 308: #1-25 odd (on 17 and 21 use trapezoids instead)