Greetings, dear guests and subscribers of my channel!
Today, I would like to devote my article to the queen of sciences, namely, mathematics! As a father of two children, I constantly help them with their homework (homework), including math. The daughters at school were asked about a hundred problems for the summer, and while checking the next one, I came across an interesting paragraph in the textbook, which is named after two great mathematicians: Newton-Simpson formula.
In fact, it refers to higher mathematics, namely to the methods of numerical integration, but due to its simplicity, they pass it in the school course. With one single universal formulaNewton-Simpson, you can calculate both the areas of figures and the volumes of various bodies.
The formula looks like this:
If the volumes of bodies are calculated, then the areas of the bases and sections are taken as "b", but if the areas are calculated, then "b" is the lengths of the bases and the segment in the center.
b1 - it is the length or area of the lower base;
b2 - this is the length of the segment in the middle of the figure or the cross-sectional area in the center of the body;
b3 - it is the length or area of the upper base;
Easier with examples ...
1. Volumes
So, suppose we need to calculate the volume of a cone or pyramid. Geometry tells us that the volume of these figures is:
V = (S * h)/3, where S - base area, h - height.
According to the Newton-Simpson formula, this is represented as follows:
V = (H / 6) * (b1 + 4b2 + b3) or (N / 6) * (b1 + 4 * (b1 / 4) + 0) = H * b1 / 3.
As you can see, Simpson's formula, through transformation, turns into a standard formula studied in school. All the same can be done with a cylinder, prism or ball, as well as with truncated versions of the pyramid and cone.
In cases with a cylinder and a prism, according to the formulaNewton-Simpsonyou will have a volume formula equal to the product of the height and the base b1, and in the case of a ball, you get the real formula for finding the volume of a sphere: 4/3 * π * r³.
Already due to the fact that the formula is applicable for finding the volumes of the most famous geometric figures, it deserves to be called universal. In addition to volume, as I wrote earlier, it can also be used to calculate areas.
2. Squares
So...
The area of any arbitrary trapezoid:
S = h / 6 * (b1 + 4 (b1 + b3) / 2 + b3) = h / 2 * (b1 + b3)
Area of a triangle:
S = h / 6 * (b1 + 4 (b1 / 2) + 0) = 1/2 * b * h
Area of a parallelogram or regular quadrangle:
S = h / 6 * (b1 + 4b1 + b1) = b * h
Q.E.D!
The formula is very simple and interesting, if your children did not go through it at school, I think that it is worth telling and showing them.
And that's all, Roman was with you, the channel "Build for Myself" ...
All the best!