V sphere = (1 / 12) (π +2) π ² r ³
I observed the circle formula is derived and found that the proportion with a circle square transformation π. I also observe the sphere, the sphere is rotated by a semicircle of a circle around the whole get up, so the round side with at least two conversion ratio, that is π square. So I derived the above formula. My formula is approximately: 1.34607088πr cube. Textbook formula is: 4/3πr cubic meters, equivalent to about 1.33333333πr cube. Numerical error is not, measurement is difficult to measure them. But these two formulas look very different, why is this so?
Linear system rotation while only 3 species: | vertical line, ---- flat-line, ╲ slash. Straight line solid body, only three kinds of rotation: cone, frustum of a cone, cylinder.
This was, GM's rotating body surface area calculation method:
s rotary table = L 2π (_r) ((_r) is the radius of average)
This formula means that the rotating body surface area is equal to rotation length multiplied by Bian Bian (which constitutes a point of rotating the edge of each rotating speed of the week averages)
Because the speed of one revolution every point is 2πr, is only connected to the rotation radius r due to the edge of different points and different. Therefore, the calculation is the key to demand r average.
Here is proof of this formula method:
Cylinder to rotate the upper and lower bottom is a radius of the edge to center of the circle as a rotary axis for rotation. Radius r is also playing this side of the radius of rotation. Circle around the rotation speed of the week 2πr. The week centered on the speed of rotation is 0. The composition of rotating edge radius r of each point is an integral point. All points, an average speed of rotation r is equal to the speed of one revolution of the mid-point. Therefore, the various points of the average speed of rotation of πr. The rotating side length r. Then, s rotary table = r πr = πr ². This is in fact a round face area.
Cylinder side area for rotating the high side of L, which constitutes the edge rotation speed of one revolution every point and so want to. Their values are as 2πr. Then, s = L 2πr.
Tapered side area in order to hypotenuse side of L as a rotary. Edges which form the rotation speed of one revolution every point with the bottom radius of a composition of each point of rotation speed is one to one week. Therefore, its average speed is equivalent to rotating a round face and underside of the average speed of rotation, namely πr. Then, s spin-side = L πr
This calculation method for the round the side of Taiwan still apply.
_r = (R-r) / 2 + r = (R + r) / 2
So: s Yuan Taiwan side = L 2π × (R + r) / 2 = πL (R + r)
Rotating edge of a round face and flat-line, cylindrical lateral side of the vertical rotation, cone lateral side of the oblique rotation. Constitutes a linear system rotation on the edge of the only three kinds. Therefore, the formula for the edge of a straight line as a rotary system is fully applicable. We are now using the mathematical basis is a natural series. Natural Sequences belonging to both the amount of arithmetic linear row cloth system. Therefore, any non-linear system of measurement must be first converted into a linear system for processing. Therefore, this method of calculation is a common rotating body surface area method of calculation.
A straight line as a rotary side, and only have a radius, on average, for rotating a curve on the edge of the radius of at least two or more average. Vertical circle radius on the average of: (1 / 4) πr circle radius on the horizontal average of: (1 / 2) r
Derived from a straight line cutting the points is still a straight line, whether horizontal or vertical cutting slitting all in the same point, the longitudinal slitting and cross-cutting point of N-point M is the same one point. Also look at sub-cut point curve analysis. Cutting the curve into the curve of the point remains. Longitudinal slitting and cross-cutting point of N-point M not on the same point. Vertical and horizontal cutting point than cutting-point higher. This is calculated with the above results reflected the situation exactly the same. That is (1 / 4) πr ratio (1 / 2) r large. But the real average is in fact the mid-point on these two points. Therefore, taking the average of these two get a real curve-off point averages.
Thus, a round rotating edge TK corresponding to the average radius of rotation is: Integrated _r = 〔(1 / 4) πr + (1 / 2) r〕 / 2 = (1 / 8) (π +2) r
Then, S Ball = (1 / 2) × 2πr × 2π_r = (1 / 4) (π +2) π ² r ²
V sphere = (1 / 3) r × (1 / 4) (π +2) π ² r ² = (1 / 12) (π +2) π ² r ³
Does the average radius of only 2? The average radius is then averaged with the arithmetic mean?
I have provided a "universal rotation of the body surface area calculation" is correct. Because that is just and textbooks on the method for calculating exactly the same as another model of equivalent transformation. This calculation method for rotating the edge of change in the slope of any two points are the uniform system, are applicable. That is, for a straight line or smooth curve with rotating side to do, apply. Have doubts about a friend, you can carefully look at that part of the contents of their own and then verify. However, in order to use this method to analyze the system curve as a rotary side when the problem emerged. Because I found that the average radius of the curve is not the only one, but two. Rotating sphere is half of the arc a week to get. This half-arc split into two, is the analysis of half a sphere. Half of the sphere is half of the vertical cross-section round. To analyze the radius, on average, to take half of this half-circle is the analysis of 1 / 4 a round face.
This 1 / 4 a round face and slightly divided into r copies, this 1 / 4 a round face and s = radius of the area of the average _r × r. So you can get a mean radius _r = (1 / 4) πr ² / r = (1 / 4) πr
This rotation side is 1 / 4 arc, to regard it as a cone rotating side view, the same micro-divided into r copies. Then, when the cone and analysis, he should be the radius of the average of (1 / 2) r.
In this way, strange things have emerged. A straight line to rotate only one side of the average radius of the curve to do at least two rotating side of the average radius of. Why is this so? This is an issue which seems to imply that, a point on the curve is likely to bring more than two pairs of coordinates. So I came up with a way to verify this issue. I have an inside or outside the arc and then draw a concentric arcs, with a ring to represent a circular arc. Clamped by two parallel straight lines parallel to a straight line with a straight line, said. Requirements, a straight line with the width and ring width is the total equal. Using the straight-line band and a straight line with the intersection, the intersection point of linear amplification analysis of the situation. Intersects with the use of ring and linear, amplification analysis of the intersection of curves and straight lines. I find that a straight line intersects with the line, whether horizontal or vertical, the intersection is always the same position. But the curves and straight lines intersect, the intersection is not in the same position. It is evident that the curve point is that with two X-coordinate, the two longitudinal coordinates, two slope. That is, a point on the curve is equivalent to two points on a straight line. This matter can also be on the other hand to understand.
Curve is continuous, if the curve point with only one slope, together with any adjacent two points are the same slope. Well, this will simply be connected together in a straight line will not be a curve. If the point is that the curve with two slopes, then any three adjacent are connected together with a curvature of the. Well, this is obviously the curve, rather than a straight line.
We are currently using analytic geometry, curves, only one point on the horizontal axis, a vertical axis. I have found the same point with more than two pairs of coordinates, and this is clearly a new analytic geometry learning. I want to know is that this point with multi-coordinate value of the analytic geometry of how the calculation, as I asked the same as above. Another problem is that the curve a little, how many pairs of coordinates. I hope that like to dig a friend, you can look at.
I raised this point with double coordinates lattice system is both feasible, but also practical. For the convenience of the people understand, I can give a simple example shows. If you use a little bit static space that the existence of a static position, then the dynamic would have little room to use two static positions, said his presence. That is, stationary point is occupied by only one position, the object is not here, where there is not here. However, a moving point is that there are both in there, not only here, there are also occupying two positions. In this way, static point, regardless of excessive length of time, is still there, so that is static. With two static spatial point the movement, then, no matter how much time is short, the movement that would come some distance, so is the movement. (Note: Where there is a point in space is bound to exist in time. So for the existence of points in space, his presence no matter how small the time can not be 0. When he was the existence of time for the most hours, that only takes a little time, that is, it only takes up one unit of time when the location of moving objects occupy a row of two geostationary space locations.) to use my method of such an analysis, you can clearly see the moving object and stationary objects the difference between the data. We are now using the displacement method is a static spatial point to correspond to the location of moving objects. Such an approach would lead to "Flight of the arrow does not move" fallacy. And my method will not be such problems.