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A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord.
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;
Parabola (magenta) and line (lower light blue) including a chord (blue). The area enclosed between them is in pink. The chord itself ends at the points where the line intersects the parabola. The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram that surrounds it.
Archimedes (died c. 212 BC) is known to have studied conics, having determined the area bounded by a parabola and a chord in Quadrature of the Parabola. His main interest was in terms of measuring areas and volumes of figures related to the conics and part of this work survives in his book on the solids of revolution of conics, On Conoids and ...
The area of the triangle ABC is exactly three times the area bounded by the parabola and the secant line AB. Proof: [1]: 15–18 Let D be the midpoint of AC. Construct a line segment JB through D, where the distance from J to D is equal to the distance from B to D. We will think of the segment JB as a "lever" with D as its fulcrum. [3]
In The Method of Mechanical Theorems he describes, for example, calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines.
Archimedes in his The Quadrature of the Parabola used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. Archimedes' theorem states that the total area under the parabola is 4 / 3 of the area of the blue triangle. His method was to dissect the area into infinite triangles as shown in the ...
This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle, the surface area and volume of a sphere, area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a ...