Mean Value Theorem for integrals

According to a mean value theorem for integrals of the continuous functions, there is a mean vahie point r G for which... 

From the mean value theorem for integration comes... 

In the multiphase flow literature these integral terms are generally approximated as the product of the interfacial area concentration and a mean interfacial flux using the mean value theorem for integrals. 

Figure A.3 The mean-value theorem for integrals states that the area under the curve/(r) from a to fo is equal to the area of a rectangle of width w = (b-a) and height/ ,. This requires that the two shaded areas be equal. The height/ , is called the mean value off(x) on the interval [a, b].

Prom the Mean Value Theorem for integrals (Section 9.14.2, p. 276),... 

This result is known as the Mean Value Theorem for integrals. 

The restoration of autonomy relies on our ability to write the integrals in Eq. (7.11) as functions of Ni. This can be done by using the mean-value theorem. For example, the second integral corresponds to the following equality ... 

In principle this integral could be applied directly to the Maxwell model to predict the decay of stress at any point in time. We can simplify this further with an additional assumption that is experimentally verified, i.e. that the function in the integral is continuous. The first value for the mean theorem for integrals states that if a function f(x) is continuous between the limits a and b there exists a value f(q) such that... 

In an attempt to find an exact formula for the integral, we may resort to the mean value theorem of calculus. This theorem states that if the integrand is evaluated at a particular known instant t = t between tn and tn+i, the integral is equal to f(T,tp(Ty)At. However, in the present case the theorem is of little use since the instant r is unknown. 

For the asymptotic stage of bubble growth, the inertial terms of the equations of motion are neglected, and the integral in Eq. (59) is simplified by physical arguments and application of the mean-value theorem to give... 

Changes in thermod)mamic properties are computed by performing integrations in some situations, the integration amounts to the evaluation of the mean for a continuous function. This mean is defined by the mean value theorem. If f x) is piecewise continuous on [a, b], then there is some value of/, designate it by/, such that... 

The surfaces Sc stand for the sides of the balance element through which a convective flow can pass. The integral terms of mass transfer can be approximated by means of Cauchy s mean value theorem in a first approximation using calculated mean values of the conserved quantities on the control volume surfaces. The mean surface values are usually determined by the central difference method or the upwind method from the function values in the control volume midpoint [32-34] ... 

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