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  2. Mean of a function - Wikipedia

    en.wikipedia.org/wiki/Mean_of_a_function

    In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by: [1] ¯ = ().

  3. Mean value theorem - Wikipedia

    en.wikipedia.org/wiki/Mean_value_theorem

    Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. [ 6 ] [ 7 ] It states: if the functions f {\displaystyle f} and g {\displaystyle g} are both continuous on the closed interval [ a , b ] {\displaystyle [a,b]} and differentiable on the open interval ( a , b ) {\displaystyle ...

  4. Gaussian integral - Wikipedia

    en.wikipedia.org/wiki/Gaussian_integral

    A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.

  5. Cauchy's integral formula - Wikipedia

    en.wikipedia.org/wiki/Cauchy's_integral_formula

    The formula can also be used to derive Gauss's Mean-Value Theorem, which states [2] = (+). In other words, the average value of f over the circle centered at z with radius r is f(z). This can be calculated directly via a parametrization of the circle.

  6. Integral - Wikipedia

    en.wikipedia.org/wiki/Integral

    A surface integral generalizes double integrals to integration over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. The function to be integrated may be a scalar field or a vector field. The value of the surface integral is the sum of the field at all points on the surface.

  7. Expected value - Wikipedia

    en.wikipedia.org/wiki/Expected_value

    Despite the newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of X is defined via weighted averages of approximations of X which take on finitely many values ...

  8. Variance - Wikipedia

    en.wikipedia.org/wiki/Variance

    where is the average value. That is, = =. The ... In these formulas, the integrals with respect to and () are Lebesgue and Lebesgue–Stieltjes ...

  9. Trapezoidal rule - Wikipedia

    en.wikipedia.org/wiki/Trapezoidal_rule

    The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable ...