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Archimedes used the method of exhaustion to compute the area inside a circle. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a sequence of polygons with an increasing number of sides and a corresponding increase in area.
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.
Archimedes' idea is to use the law of the lever to determine the areas of figures from the known center of mass of other figures. [1]: 8 The simplest example in modern language is the area of the parabola. A modern approach would be to find this area by calculating the integral
The area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere. The area of a segment of a parabola determined by a straight line cutting it is 4/3 the area of a triangle inscribed in this segment. For the proofs of these results, Archimedes used the method of exhaustion attributed to ...
The law of demand, however, only makes a qualitative statement in the sense that it describes the direction of change in the amount of quantity demanded but not the magnitude of change. The law of demand is represented by a graph called the demand curve, with quantity demanded on the x-axis and price on the y-axis. Demand curves are downward ...
Archimedes' achievements in this area include a proof of the law of the lever, [10] the widespread use of the concept of center of gravity, [11] and the enunciation of the law of buoyancy known as Archimedes' principle. [12] In astronomy, he made measurements of the apparent diameter of the Sun and the size of the universe.
The lever and its properties were already well known before the time of Archimedes, and he was not the first to provide an analysis of the principle involved. [5] The earlier Mechanical Problems, once attributed to Aristotle but most likely written by one of his successors, contains a loose proof of the law of the lever without employing the concept of centre of gravity.
The shift of a demand curve takes place when there is a change in any non-price determinant of demand, resulting in a new demand curve. [11] Non-price determinants of demand are those things that will cause demand to change even if prices remain the same—in other words, the things whose changes might cause a consumer to buy more or less of a ...