Using the midpoint method, and taking symmetry into account, develop an efficient algorithm for

step 1 Hello, number eight, we're back. Okay, welcome back. So we're chapter six, section three of Stew

Using the midpoint method, and taking symmetry into account,...

Key Concepts

Exploiting Symmetry

Exploiting symmetry involves recognizing and using the inherent symmetrical properties of a curve to reduce computational work. By calculating the pixel points for one portion of a symmetric curve and then mirroring these points to other portions, the algorithm maximizes efficiency, minimizes redundant calculations, and simplifies the overall scan conversion process.

Scan Conversion

Scan conversion refers to the process of converting a mathematical or geometric description of a curve into a set of discrete points (pixels) that can be displayed on a raster device. This process involves determining which pixels best approximate the continuous curve, making it a fundamental aspect of rendering in computer graphics.

Midpoint Method

The midpoint method is an incremental algorithm used to determine the next pixel in drawing a curve by evaluating the position of the midpoint between two candidate pixels relative to the ideal curve. This approach allows for efficient decision-making without heavy reliance on floating-point arithmetic, making it both fast and accurate for generating digital approximations of continuous curves.

Transcript

00:01 Hello, number eight, we're back.

00:02 Okay, welcome back.

00:03 So we're chapter six, section three of stewart's calculus of early transcendental seventh edition, page 445, number 27.

00:16 So we're given the square root of a one plus x cubed, you graph that on your favorite graphing utility.

00:25 And we're interested only from x equals zero to one.

00:30 So it's something like this.

00:31 If this is x equals 1 and this is y equals 1 not to scale so this is the region we're revolving about the y axis here's the region and so we have to have vertical strips we're doing cylindrical shells is asked to do so the thickness is the x the height is the function y equals rad 1 plus x cubed um and the radius interpretation is just x away the strip is x away from the y axis so i'm going to say v is 2 pi the integral from 0 to 1 x times the square root of 1 plus x cubed the x all right that's an x cubed doesn't look like it is all right and we can punch this up in the calculator and get a numerical estimate board what they want you to do is figure out an estimate using midpoint formulas let's see using n equals 5 so five remand rectangles are using the height from the midpoint so we're chopping up the regions of five equal pieces right so delta x dot the x is going to be b minus a over n which is b is one end of the integrals limit of integration the other is a is zero and then the number of rectangles is five so this is one fifth of point five five right now so now let's try let's think of this as like an area problem if i graph the region in question let's graph it go to y equals let me graph two pi x you're gonna do it sorry i got to do a plots off here all right so y equals let's see what this looks like two pi x square root of one plus x cubed okay i'm gonna do zoom four i'm only interested from zero to one all right so let me trace this thing it goes to zero zero and it goes to one comma eight point eight eight six and it's pretty steep to see what it looks like i'm not going to draw it to scale let's see let's draw this out a little bit and i'm just going to draw it to scale i'm just going to get visualized what the rectangles might look like right so one -fifth two -fits three -fits four fifths five -fifths and it goes up to eightish eight point what is it eight eight eight six now so one and eight eight eight six eight eight eight six is not the scale it's pretty steep let's say it's doing like this i don't actually it's slightly kind of cable up so say like that so here's one based from one to one fifth and another base base of a re -one rectangle from one -fifth to two -fifths, another base from two -fits to three -fifths, and from three to four -fits, and from four to five -fifths.

04:56 But we're taking the heights halfway in between.

05:02 So there's your first height, at this height, at this height, at this height, at this height, at this height.

05:15 So the first rectangle is this tall, let's say, the next one is this tall, the next one is this tall, next one is this tall, next one is this tall.

05:26 So if that's one fifth, we're halfway in between.

05:37 Let's see.

05:39 So i guess i would mean that we're at, if this is point two, this is point one.

05:48 If this is point two, and point four, this is point three, correct? so f of point three, f of point five, f of point seven, f of point nine.

06:05 So the volume is going to be approximated by if the function is 2 pi x times a square to 1 plus x cubed, it's going to be delta x, which is 0 .2 or 1 5th, 0 .2 times the function at 0 .1, plus the function of 0 .2, plus the function of 0 .3, plus the function of 0 .5, plus the function of 0 .7, plus the function, at 0 .9.

06:54 Now i put f, which is 2x, square root of x was 1 plus x cubed, into my y equals.

07:02 So i'm going to type this all out of my calculator exactly as is, but instead of f, i'm going to use y1...

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