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For instance, in the latter, Simpson's 3rd rule is used to find the volume between two co-ordinates. To calculate the entire area / volume, Simpson's first rule is used. [7] Simpson's rules are used by a ship's officers to check that the area under the ship's GZ curve complies with IMO stability criteria.
The area under the effect curve (AUEC) is an integral of the effect of a drug over time, estimated as a previously-established function of concentration. It was proposed to be used instead of AUC in animal-to-human dose translation, as computer simulation shows that it could cope better with half-life and dosing schedule variations than AUC.
The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). [43]
The area of the surface of a sphere is equal to quadruple the area of a great circle of this sphere. The area of a segment of the parabola cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment. For the proof of the results Archimedes used the Method of exhaustion of Eudoxus.
While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not necessarily just rectangles, and so it is more flexible. For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions.
For reversible (ideal) processes, the area under the T–s curve of a process is the heat transferred to the system during that process. [1] Working fluids are often categorized on the basis of the shape of their T–s diagram. An isentropic process is depicted as a vertical line on a T–s diagram, whereas an isothermal process is a horizontal ...
The term "quadrature" is a traditional term for area; the integral is geometrically interpreted as the area under the curve y = x n. Traditionally important cases are y = x 2, the quadrature of the parabola, known in antiquity, and y = 1/x, the quadrature of the hyperbola, whose value is a logarithm.
The area under the curve (often referred to as simply the AUC) is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one (assuming 'positive' ranks higher than 'negative'). [26]