Search results
Results From The WOW.Com Content Network
Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean. [1] Define sequences a n and g n, where a 0 = 1, g 0 = √ 1 − k 2 = k ′ and the recurrence relations a n + 1 = a n + g n / 2 , g n + 1 = √ a n g n hold.
An ellipse has a simple algebraic solution for its area, but for its perimeter (also known as circumference), integration is required to obtain an exact solution. Analytically , the equation of a standard ellipse centered at the origin with width 2 a {\displaystyle 2a} and height 2 b {\displaystyle 2b} is: x 2 a 2 + y 2 b 2 = 1. {\displaystyle ...
The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height (the quadrature of the parabola); The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes;
The area of a two-dimensional region can be calculated using the aforementioned definite integral. [50] The volume of a three-dimensional object such as a disc or washer can be computed by disc integration using the equation for the volume of a cylinder, , where is the radius.
The square gets sent to a rectangle circumscribing the ellipse. The ratio of the area of the circle to the square is π /4, which means the ratio of the ellipse to the rectangle is also π /4. Suppose a and b are the lengths of the major and minor axes of the ellipse. Since the area of the rectangle is ab, the area of the ellipse is π ab/4.
In more recent years, computer programs have been used to find and calculate more precise approximations of the perimeter of an ellipse. In an online video about the perimeter of an ellipse, recreational mathematician and YouTuber Matt Parker, using a computer program, calculated numerous approximations for the perimeter of an ellipse. [10]
The incomplete elliptic integral of the first kind is defined as, (,) = (),the second kind as (,) = (),and the third kind as (,,) = ( ()) ().The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign.
where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b. = ... (when the path of integration winds once counterclockwise around 0.