Zero Integrals: The Theory Behind Evaluating Expressions

[hasanhabibul]

what is the theory behind ...if close integral of a expression is zero..then the expression is itself zero

 

[Born2bwire]

Do you mean that if we have an integral whose path is a closed loop that is equal to zero then the integrand must be zero? If that is so then we cannot say that it is true in general, there need to be addtional conditions. For example, the integral of a conservative force over a closed loop will be zero but the integrand, the force, does not necessarily need to be zero itself over the integration path.

I cannot remember all of the exact conditions where it follows that the integrand has to be identically zero. One example is the Euler-Lagrange equations, in their derivation we can come to the point where,

[tex] \int_0^1 F(x)\eta(x)dx = 0 [/tex]

Here, the function \eta(x) is any arbitrary function that enforces the boundary conditions \eta(0)=\eta(1)=0. Since \eta(x) is arbitrary between the endpoints of the path then we conclude that the integrand, F(x), must be identically zero. This is not the exact case that you requested since it is not a closed path but one could extend the same thinking to a closed path.

That is, if we have an integrand that is the product of two functions, one that is an arbitrary function, then for the integral to be zero the the non-arbitrary function must be zero over the entire integration path. Thus, the integrand itself must be zero.

 

[charng]

The theory behind zero integrals is based on the fundamental theorem of calculus, which states that the integral of a function can be evaluated by finding its antiderivative and evaluating it at the boundaries of integration. In other words, the integral of a function represents the area under its curve, and if the area is zero, then the function itself must be zero.

When evaluating an integral, we are essentially finding the net change of the function over a given interval. If the net change is zero, it means that the function is neither increasing nor decreasing over that interval, and therefore must be constant. This constant value is then evaluated at the boundaries of integration, resulting in a zero integral.

In the case of close integrals, where the integral is taken over a closed interval, the boundaries of integration are the same, and therefore the function must be zero over that interval. This is because if the function had any non-zero values, the net change over the interval would not be zero, and the integral would not be zero.

In summary, the theory behind zero integrals is that if the integral of a function is zero, then the function itself must be zero. This is based on the fundamental theorem of calculus and the concept of net change over an interval.