The upper Riemann integral is the largest possible value that the integral can take, while the lower Riemann integral is the smallest possible value. In other words, the upper Riemann integral is the supremum of all possible sums, while the lower Riemann integral is the infimum.
The Riemann-Stieltjes integral is a generalization of the Riemann integral, where the integral is taken with respect to a general function instead of just the variable x. The upper and lower Riemann integrals are used to define the Riemann-Stieltjes integral, with the upper integral corresponding to the upper sum and the lower integral corresponding to the lower sum.
The upper and lower Riemann integrals can be calculated using the Riemann sum method, where the interval is divided into smaller subintervals and the supremum or infimum of each subinterval is taken. As the number of subintervals increases, the upper and lower sums approach the upper and lower Riemann integrals, respectively.
The upper and lower Riemann integrals are important in the study of continuous functions and their integrals. They are used to approximate the area under a curve, and can be used to determine the convergence or divergence of a series. In real-world applications, they are used in fields such as physics, engineering, and economics.
Yes, the upper and lower Riemann integrals can be equal in certain cases. This occurs when the function being integrated is constant over the interval of integration. In this case, the upper and lower sums will be equal, resulting in the same value for both the upper and lower Riemann integrals.
