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Okay so this week we only really learned stuff on Wednesday and Thursday..And it was for that UNO study. So first we reviewed Sigma notation and how to evaluate the sums and whatnot. Then we went over the different methods one can use to find the area of a region bounded by a curve and the x-axis--You can use Riemann sums (LRAM & RRAM)-(break up the region into shapes, find the area of each shape and then add them all together); and you can use the Fundamental Theorem of Calculus (which is when you take the integral of the function and then use the formula).
Here's an example:

1.) Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.
f(x) = 2x+5 [0,2]; 4 rectangles
*Well you're going to need to find LRAM and RRAM using only 4 rectangles so the first thing you have to figure out is the width of each rectangle. To do this you can use a little formula where you subtract the numbers of the interval over the number of rectangles, which would be this:
2-0/4 and that gives you a width (delta x) of 1/2
*Now you have to plug in the numbers between 0 and 2 into the original equation. (The numbers you're going to plug in are 0, 1/2, 1, 3/2, and 2...because you add 1/2 each time and there's 4 rectangles)
*So here's what you get when you plug in those numbers:
2(0)+5 = 5
2(1/2)+5 = 6
2(1)+5 = 7
2(3/2)+5 = 8
2(2)+5 = 9
*Now you need to find LRAM & RRAM. So first, let's start with LRAM. To find this, you're going to start with the numbers to the left (starting with 0) and you're NOT going to plug in the last number on the right. And you're going to multiply each bolded number by 1/2 because that's the width of each shape, and then you'll add them all together to get the area
*So for LRAM you should have this:
1/2(5)+1/2(6)+1/2(7)+1/2(8)
=13 (which is an underestimate)
*And for RRAM you should have this:
1/2(6)+1/2(7)+1/2(8)+1/2(9)
= 15 (which is an overestimate)..and you don't plug in what you got for 0 because you're looking at values from the right
*And if you go on to use the Fundamental Theorem of Calculus you get that the area is 14, which is right in the middle of your estimates

*I'm not sure if I explained this all that well, but I'm just getting the hang of it haha..