1/30/11

Alrightttt, I think this week was the last week of that UNO program. The material really wasn't hard to understand at all. We covered LRAM,RRAM,MRAM,TRAM, finding areas between regions, disks&washers, and lastly we learned how to find the volume. I would explain volume since it's the most recent thing we've learned but since I don't have my notes I'll explain how to find the area between two curves.

AREA BETWEEN TWO CURVES:
This is when you're given two separate equations: you first graph each equation, then you have to decide which curve is on top, and which one is on the bottom(this will help when plugging into the formula, so you must graph them). After deciding which equation is the top, or bottom you use this formula: S top-bottom (where you subtract the two equations and then integrate it). After that (if you aren't given the bounds in the directions), to find them all you do is set the two equations equal to each other and solve for x. Then once you integrate(Don't forget there is a formula to integrate if you don't know how to do it :)), you plug in the numbers of the bounds into the area formula (Fundamental Theorem of Calculus)->f(b)-f(a). That's ittttt!

So here's an example:

1.) Find the area between the two curves for y = x^2 + 2x + 1 and y = 2x + 5.
*Okay so the first thing you have to do is draw this graph so that you can figure out which one of these curves is on top.
*So the graph of x^2+2x+1 is a parabola with vertex at x=-1 opening up...And the graph of 2x+5 is a line going up intersecting the parabola at points (-5/2,0) and (0,5). So that means that the line is on top and the parabola is on the bottom.
*Before we find the area between the two graphs, find the bounds since we were not given them...So to do that you just set your two equations equal to each other and solve for x like this:
x^2+2x+1 = 2x+5
x^2-4
(x+2)(x-2)
x=2, -2 <-that's the bounds: 2 going on the top, and -2 on the bottom
*Now we can use the formula to find the area. We already said that the line 2x+5 is on top so here's how to set up the equation:
S(2x+5)-(x^2+2x+1)
Now distribute the negative and simplify it to get this:
S -x^2+4
Now integrate that and you should end up with this:
-1/3x^3+4x
Now you can use the Fundamental Theorem of Calculus to plug in your bounds and find the area. So this is what you should get when you plug in:
-1/3(2)^3+4(2)-[-1/3(-2)^3+4(-2)]
= 32/3 <-AREA

That's about all there is too it! I think I understood everything we learned pretty well.. Now I guess were gonna start the AP stuff... ahhhh :\