Do you mean the Reimann integral? For each value that they are not equal consider a small interval (small enough that only one point of inequality is included). Then |Sup(f)-Sup(g)|=|f(x*)-g(x*)|>0. Then consider the effect of all the points of inequallity on the upper integrals, then likewise for the lower integrals.jeanf said:i don't know where to start on this problem. could someone help me please? thanks.
let [tex] f: A -> R [/tex] be an integrable function, where A is a rectangle. If g = f at all but a finite number of points, show that g is integrable and [tex] \int_{A}f = \int_{A}g. [/tex]
The concept of "equality of integrals" refers to the idea that two integrals with the same limits of integration and integrands will have the same value.
The equality of integrals can be proven by using the fundamental theorem of calculus, which states that if the integrand is continuous on a closed interval, then the integral is equal to the antiderivative evaluated at the endpoints of the interval.
No, two integrals with different integrands will not be equal, even if they have the same limits of integration. The integrand is a crucial factor in determining the value of an integral.
The concept of "equality of integrals" is essential in many areas of mathematics, such as calculus, differential equations, and physics. It allows for the evaluation of complicated functions and is the basis for many other mathematical concepts.
Yes, there are some exceptions to the rule of "equality of integrals." For example, improper integrals, where the limits of integration are infinite or the integrand is undefined, may not follow this rule. Additionally, some functions may have different antiderivatives, leading to different values for the integral.