By Brehm U., Kuhnel W.
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Additional info for 15-vertex triangulations of an 8-manifold
In that case, the notation ∫f(x) + dx may be used for an antiderivative of f, with the understanding that an arbitrary constant can be added to any antiderivative. Thus, ∫f(x) dx + c, 50 where c is an arbitrary constant, is an expression that gives all the antiderivatives. antidifferentiate The reverse process to *differentiation used as a method of *integration when not working from first principles as the limit of a sum of incremental elements. antilog Abbreviation for *antilogarithm. antilogarithm The antilogarithm of x, denoted by antilog x, is the number whose *logarithm is equal to x.
The areas can be calculated by simple formulae based on their dimensions. Other more complex plane figures or 3-dimensional surfaces require the use of integration, or numerical approximations to find their areas. area of a surface of revolution Let y = f(x) be the graph of a function f such that f′ is *continuous on [a, b] and f(x) ≥ 0 for all x in [a, b]. The area of the surface obtained by rotating, through one revolution about the x-axis, the arc of the curve y = f(x) between x = a and x = b, equals Parametric form For the curve x = x(t), y = y(t)(t ∈ [α, β]), the surface area equals Polar form For the curve r = r(θ)(α ≤ θ ≤ β), the surface area equals 59 area under a curve Suppose that the curve y = f(x) lies above the x-axis, so that f(x) ≥ 0 for all x in [a, b].
Ryabhata (about 476-550) Indian mathematician, author of one of the oldest Indian mathematical texts. Written in verse, the Āryabhatīya is a summary of miscellaneous rules for calculation and mensuration. It deals with, for example, the areas of certain plane figures, values for π, and the summation of *arithmetic series.