00:04
So problem 5 .8, we're dealing with area under curves.
00:08
We're asked to find the upper and lower sums as estimates for the area for the number of rectangles being three and four.
00:18
So real roughly, this is a parabola that opens upward.
00:22
So if i think about this, and it's been shifted vertically up one.
00:27
So the graph of this guy is going to look like this.
00:33
And so we're interested in finding estimates for this area that you see that's all shaded.
00:43
Okay.
00:44
So what i notice is the curve is decreasing from negative 1 to 0, and it's increasing from 0 to 1.
00:53
So that means it's going to be a difference in terms of when i look at upper sums versus lower sums, and i'm going to have to shift which side of the rectangle i'm using for the heights in order to make that happen.
01:03
So that's the rough curve.
01:05
Let's look at something a little bit more precise.
01:07
So if i look at this, number of intervals equal three.
01:11
So i'm going from negative one to one on the x -axis.
01:14
And if i use three rectangles.
01:16
What you can see is if i start out, if i use the left side as the height, then that first rectangle, that's an upper sum.
01:24
And if i look at the next one, if i use, it's going to be either way.
01:28
The left or the right, i still get an upper because the rectangle is above the curve.
01:34
And then you can see that, well, that last section there in the last rectangle, i cannot use the left sum.
01:40
I'm going to have to use the right sum, maybe the right end point of the rectangle in order to make that an upper sum.
01:47
So if i'm using n equal three, then how is this going to work? so the distance from negative one to one, so the width of the rectangle is going to be one minus negative one, divide that by three.
02:05
So that is two -thirds is the width of the rectangle.
02:11
So for the n, so let's look at the cases that we have, we're going to have n equal three and n equal four.
02:21
We're asked to come up with an upper sum and a lower sum.
02:30
So to find the upper sum, when n is equal to three, so you're going to see it's going to be two -thirds is the width of each, rectangle.
02:41
So two -thirds, and then that first rectangle is going to be the value of the function at negative 1.
02:46
So this is going to be two -thirds, f of negative 1 plus.
02:57
And then the next rectangle is going to be, it doesn't matter, since it is an even function, f of negative 1 -third or f of 1 -third, so plus f of 1 -third.
03:10
And then that last rectangle, instead of using a left endpoint, it's going to have to be the right end point so f of one and if we evaluate this what do we come up with f of negative one is two so this number is two and then also this number right here is two f of one -third that's one -ninth plus one that is going to be 10 ninths and so this is going to be if i look at that real quick so four plus ten ninths so i think that's four thirty six nine so this is two -thirds times forty six over nine which is twenty seven over ninety two yes ninety two so that's my upper sum using three now if you look at the lower sum go back to the curve here in order to get a lower i'm just change which side of the rectangle that i'm looking at...