Scribe Post 3/7

   Today Mr. O’Brien started class by reminding us that only two of us had signed up for the AP test and the rest of us should sign up soon. Then we got our supercorrection follow-up tests back and went over the problems to make sure we fully understood each one.
Follow-up test:
1. Use fnInt!
2. Use trapezoidal rule and remember the area of a trapezoid:
3. Average velocity:
Easy mistake to make is forgetting to multiply by the fraction in front.
4. a. Find acceleration using nDeriv
   b. Total distance:
Absolute value around the function because distance cannot be negative so it makes the part of the     graph below the x axis positive.
   c. Position at t=5: use the fundamental theorem of calculus.
5. Find the antiderivative. Don't forget to add C. Check to make sure you are right by taking the derivative.
6. Find antiderivative of the function then subtract the antiderivative evaluated at a from the antiderivative evaluated at b.
7. Sketch a graph of the absolute value of x shifted to the right two and up one. Then either form two trapezoids and sum their areas or count squares.
8. Separate the integral then use the graph to integrate each and add them together.
9. Integration rules.

   After all remaining questions about the test were answered Mr. O'Brien informed us that after class on Monday we would be done with calculus! After Monday we will review up until the AP test. Before we moved onto an example problem of the new stuff we looked at number 47 from IW 1.
IW1:
47) u substitution problem.


Area between curves:

Then we moved on to look at the first part of the new section by staring a free response question from the 2000 AP exam. In this last section we are going to be finding the area of anything whose boundary is determined by a mathematical curve. After this section we will have a way to calculate the area and the volume of objects in the real world.

This question asks us to find the area of region R which is bound by two curves.
First use a representative rectangle which is a rectangle like you would use to find a Riemann sum. Then find a function for the height of the rectangle and find a base.
Base: change in x which is known as dx.
Height: the difference between the top point and the bottom point.


Now we sum the height times the dx from 0 to 1 and we get this integral.
Now we can use fnInt to solve this integral, giving us the area of region R.  
As long as you subtract the top from the bottom it doesn't matter what quadrant you are in, the areas between the curves will always be positive.

IW3 pg 399 Example problem for finding the area between two curves.


2) Find area of the shaded region. No technology. 
 
We do this problem the same way we did the free response problem. Find the height of a representative rectangle by subtracting top from bottom and integrating. Once we find our height function we multiply it by the base and integrate to find the area:  

This video gives a clear explanation of how to find the area between two curves as well as a demonstration on how to do it on your calculator.

IW 3 p. 399/2, 4, 10, 14
p. 401/3, 5, 13, 33, 39, 47, 49, 52, 53, 54, 55
Don't forget to post questions!
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