For each of the three graphs: a. Obtain a lower bound L(A) for the area enclosed by the curve by

For each of the three graphs: a. Obtain a lower bound L(A)...

Key Concepts

Bounding Techniques

Bounding techniques in area estimation involve determining both lower and upper estimates to confine the true area within a certain range. This strategy is central in numerical integration and approximation methods, ensuring that the estimated value is as close as possible to the true area by capturing the uncertainty introduced by partial coverage.

Upper Bound Estimation

Upper bound estimation is achieved by adding the areas of the partially enclosed squares to the lower bound value. This yields a guaranteed overestimate of the area, offering an interval within which the true area lies.

Grid Approximation

This is a method where a grid is overlaid on a continuous region and the area is estimated by summing the areas of the squares that the grid creates. The approach discretizes a complex, often irregular, region into simple, uniform shapes, making it easier to estimate the area using basic arithmetic.

Lower Bound Estimation

Lower bound estimation involves summing up the areas of all grid squares that are completely contained within the region. This technique provides a guaranteed underestimate of the actual area since it only includes parts of the region that are fully covered by a square.

Transcript

00:01 Okay, for problem 57, we are going to try to find the lower bound for the area of this picture that's enclosed by this picture.

00:16 And all that means is, you know, the least amount of area that is included by all of the full squares.

00:27 Each of these squares are one by one.

00:32 So all of the full squares that are enclosed in this picture.

00:37 And if we look at this, there aren't really any full squares in here.

00:44 Because like this square, a little bit of it's outside the picture.

00:51 And this square, you know, a little bit it's outside the picture.

00:55 And we could go all the way around.

00:57 There's not a single square that is fully inside this picture.

01:01 So that means we can't use any.

01:03 Of these as an estimate for the area.

01:08 So this first one's going to be zero.

01:12 There's no full squares inside the picture.

01:14 This little b of a is the amount of squares that are partially used in the picture.

01:23 And well, there's several of those.

01:27 I could just, i'm not going to outline them all, but we can see just six of them up here, six of them up here.

01:37 Six of them up and there'd be six of them down here, then there'd be four more and four more for a total of 20.

01:49 So the upper bound area includes, that's part b, the upper boundary includes the lower bound area plus this partially bound area.

02:02 And we're just going to add these together and get 20.

02:08 That's pretty easy to see from this simple picture.

02:13 Now if we go to a little bit different picture, there's just three graphs included in this problem.

02:25 The lower area is going to be all of the area completely full boxes enclosed in this picture.

02:41 Well, the problem is that this grid is not quite drawn to scale.

02:51 That each of these, this is three up here.

02:56 So each of these are going to be a half.

03:01 And each in this way is also going to be a half.

03:04 And so a half by a half is a fourth.

03:07 So each of these boxes represents a fourth.

03:11 That's okay.

03:11 We can count how many fourths there are.

03:15 One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve...

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