HSC EV Higher Mathematics 1st Paper 10th Chapter Note. Integration. This article is about the concept of definite integrals in calculus. For the indefinite integral, see antiderivative. For the set of numbers, see integer. For other uses, see Integral (disambiguation). In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral:
HSC EV Higher Mathematics 1st Paper 10th Chapter Note. Integration
The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann gave a rigorous mathematical definition of integrals. It is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function, as well as the domain over which the integration is performed, has been generalized. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space.
Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. Common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity, and day length, and average temperature variations through the year.
Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. Common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity, and day length, and average temperature variations through the year.
In mathematics, a combination is a selection of items from a collection, such that (unlike permutations) the order of selection does not matter. For example, given three fruits, say an apple, an orange, and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant can be used to solve those equations, although more efficient techniques are actually used, some of which are determinant-revealing and consist of computationally effective ways of computing the determinant itself. The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra. In analytic geometry, determinants express the signed n-dimensional volumes of n-dimensional paralleled. Sometimes, determinants are used merely as a compact notation for expressions that would otherwise be unwieldy to write down. When the entries of the matrix are taken from a field (like the real or complex numbers), it can be proven that any matrix has a unique inverse if and only if its determinant is nonzero. Various other theorems can be proved as well, including that the determinant of a product of matrices is always equal to the product of determinants; and, the determinant of a Hermitian matrix is always real.
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