Atanov's formula (PDF)

File information

This PDF 1.5 document has been generated by Microsoftรยฎ Word 2016, and has been sent on pdf-archive.com on 06/12/2015 at 15:20, from IP address 178.49.x.x. The current document download page has been viewed 527 times.
File size: 421.31 KB (2 pages).
Privacy: public file

File preview

Atanovโs formula for area of a parabolic segment
Given a parabola ๐ = โ๐๐ + ๐๐ + ๐, which is passed through by a line ๐ = ๐๐ + ๐ at points (๐๐ ; ๐๐ ) and
(๐๐ ; ๐๐ ) (Fig. 1). Then the area of parabolic segment is given by

๐จ=

(โ๐)๐
๐

, where โ๐ = ๐๐ โ ๐๐

Proof.
๐๐

๐๐

๐๐

๐จ = โซ (โ๐๐ + ๐๐ + ๐) ๐๐ โ โซ (๐๐ + ๐) ๐๐ = โซ (โ๐๐ + (๐ โ ๐)๐ + (๐ โ ๐)) ๐๐
๐๐

๐๐

๐๐

where equation ๐ = โ๐๐ + (๐ โ ๐)๐ + (๐ โ ๐) is a parabola, which intersects x-axis at points (๐๐ ; ๐) and
(๐๐ ; ๐) (Fig. 2). Thus, we see that the original area is equal to the area of a segment bounded by this
parabola and the x-axis .

If we shift the origin to point (๐๐ ; ๐) (Fig. 3), parabola will be represented by the equation ๐ = โ๐(๐ โ โ๐).

โ๐

โ๐

โ๐

๐๐ โ๐ โ ๐๐
๐จ = โซ โ๐(๐ โ โ๐) ๐๐ = โซ (โ๐ + โ๐ โ ๐) ๐๐ = [โ +
]
๐
๐
๐
๐

๐

๐

โ๐๐ โ๐ โ โ๐๐ โ๐๐
๐จ=โ
+
=
๐
๐
๐
It's quite easy to see that in more common case when parabola is represented by the equation
๐ = ๐๐๐ + ๐๐ + ๐, coefficient ๐ appears in the formula:
|๐|โ๐๐
๐จ=
๐

Atanov's_formula.pdf (PDF, 421.31 KB)

HTML Code

Copy the following HTML code to share your document on a Website or Blog