sectionsellipse

A section of an ellipse is the region bounded by an arc of the ellipse and a chord connecting the endpoints of the arc. This concept is analogous to a segment of a circle, but applied to the more general elliptical shape.

The area of a section of an ellipse can be calculated using integration. If the ellipse is defined by the equation (x^2/a^2) + (y^2/b^2) = 1, and the chord connects two points on the ellipse, the area can be found by determining the definite integral of the function representing the upper (or lower) boundary of the ellipse between the x-coordinates of the chord's endpoints, and then subtracting the area of the trapezoid (or triangle) formed by the chord and the x-axis. Alternatively, a more general approach involves parameterizing the ellipse and integrating with respect to the parameter.

There are specific formulas for common sections. For instance, a sector of an ellipse, bounded by two radii from the center to the ellipse and the arc between them, has an area proportional to the angle subtended at the center. This proportionality is similar to sectors in a circle, but the relationship between the angle and area is more complex due to the differing semi-axes.

Understanding sections of an ellipse is important in various fields, including celestial mechanics, where planetary orbits are elliptical. The study of the areas swept out by a planet's radius vector (connecting the planet to the Sun at one focus) over time, as described by Kepler's second law, involves elliptical sectors.