1. Warmup 11/2/16 Objective Tonight’s Homework
How do we know we can trust that the Bible hasn’t been changed over the last few thousand years?
Objective Tonight’s Homework
To learn how to calculate the area under a curve
pp 209: 1, 2, 4, 5, 8, 12

2. Homework Help
Let’s spend the first 10 minutes of class going over any problems with which you need help. 2

3. Notes on the Area Under a Curve
Being able to find the volume or area of something is useful.

4. Notes on the Area Under a Curve
Being able to find the volume or area of something is useful.
Let’s start our considerations with the following. Find the areas for the following shapes:
4 3

5. Notes on the Area Under a Curve
Being able to find the volume or area of something is useful.
Let’s start our considerations with the following. Find the areas for the following shapes:
4 3 π

6. Notes on the Area Under a Curve
Those shapes are pretty easy. The area under a curvy shape is much harder.
Any ideas? A = ?

7. Notes on the Area Under a Curve
Those shapes are pretty easy. The area under a curvy shape is much harder.
Any ideas? A = ?
Let’s break the area into rectangular strips!

8. Notes on the Area Under a Curve
Those shapes are pretty easy. The area under a curvy shape is much harder.
Any ideas? A = ?
Let’s break the area into rectangular strips!
But what to do about the tops? These aren’t quite rectangles yet. We have to make an estimation here.

9. Notes on the Area Under a Curve
Those shapes are pretty easy. The area under a curvy shape is much harder.
Any ideas? A = ?
Let’s break the area into rectangular strips!
But what to do about the tops? These aren’t quite rectangles yet. We have to make an estimation here.
There are multiple ways we can do this…

10. Notes on the Area Under a Curve
Lower Sum
In this method, we estimate each rectangle as a little smaller than the area in reality.

11. Notes on the Area Under a Curve
Lower Sum
In this method, we estimate each rectangle as a little smaller than the area in reality.
We call each rectangle a “partition”. The more rectangles we have, the better our estimation will be.

12. Notes on the Area Under a Curve
Lower Sum
In this method, we estimate each rectangle as a little smaller than the area in reality.
We call each rectangle a “partition”. The more rectangles we have, the better our estimation will be.
To find the area of each rectangle, multiply the base (even divisions) by the height (the y-value of the function at the x on the left side).

13. Notes on the Area Under a Curve
Upper Sum
In this method, we estimate each rectangle as a little bigger than the area in reality.

14. Notes on the Area Under a Curve
Upper Sum
In this method, we estimate each rectangle as a little bigger than the area in reality.
To find the area here, multiply the base (even divisions) by the height (the y-value of the function at the x on the right side).

15. Notes on the Area Under a Curve
If we want to find our final area, we know that it is above the lower sum and below the upper sum.
lower sum < real area < upper sum
It’s probably about the average between the two, but not exactly.

16. Notes on the Area Under a Curve
Example:
Find the area under the curve y=(x+2)2 – 1 between 0 and 4 using n = 4 partitions

17. Notes on the Area Under a Curve
Example:
Find the area under the curve y=(x+2)2 – 1 between 0 and 4 using n = 4 partitions
First we set up our graph. 35 8 3

18. Notes on the Area Under a Curve
Example:
Find the area under the curve y=(x+2)2 – 1 between 0 and 4 using n = 4 partitions
First we set up our graph.
Next we draw our lower sums. 35 8 3

19. Notes on the Area Under a Curve
Example:
Find the area under the curve y=(x+2)2 – 1 between 0 and 4 using n = 4 partitions
First we set up our graph.
Next we draw our lower sums.
Now we find the area of each and add them together.
Lower sum = (1•3)+(1•8)+(1•15)+(1•24)
= (3)+(8)+(15)+(24) = 50 35 8 3

20. Notes on the Area Under a Curve
Example:
Find the area under the curve y=(x+2)2 – 1 between 0 and 4 using n = 4 partitions
First we set up our graph.
Now we draw our upper sums. 35 8 3

21. Notes on the Area Under a Curve
Example:
Find the area under the curve y=(x+2)2 – 1 between 0 and 4 using n = 4 partitions
First we set up our graph.
Now we draw our upper sums.
Like the lower, now we just add.
Upper sum = (1•8)+(1•15)+(1•24)+(1•35)
= (8)+(15)+(24)+(35) = 82 35 8 3

22. Notes on the Area Under a Curve
So at best we can say that our answer is this:
50 < real area < 82

23. Group Practice
Look at the example problems on pages 206 and 207. Make sure the examples make sense. Work through them with a friend.
Then look at the homework tonight and see if there are any problems you think will be hard. Now is the time to ask a friend or the teacher for help!
pp 209: 1, 2, 4, 5, 8, 12 23