Integral of Limit in Riemann Sum
In order to find the integral of the limit in a Riemann sum, we need to first understand the concept of Riemann sums and integration.

Riemann Sum:
A Riemann sum is a method for approximating the area under a curve by dividing the region into smaller rectangles and summing up their areas. As the number of rectangles increases, the approximation becomes more accurate.

Integral Calculation:
To calculate the integral of the limit in a Riemann sum, we can start by finding the Riemann sum for the function 2x over the interval [1, 2]. This involves dividing the interval [1, 2] into smaller subintervals, calculating the height of each rectangle using the function 2x, and summing up the areas of all the rectangles.

Limit in Riemann Sum:
In this case, we are interested in finding the integral of the limit as the number of subintervals approaches infinity. This means that we need to take the limit of the Riemann sum as the width of the subintervals approaches zero.

Calculation:
The Riemann sum for the function 2x over the interval [1, 2] can be calculated as follows:
Σ(2xΔx) where Δx is the width of each subinterval.
Taking the limit of this sum as Δx approaches zero will give us the integral of 2x over the interval [1, 2]. This integral can be calculated using the fundamental theorem of calculus or by evaluating the antiderivative of 2x with respect to x over the interval [1, 2].
Therefore, the integral of 2x dx from 1 to 2 is equal to [x^2] evaluated from 1 to 2, which gives us the result of 4.
In conclusion, by understanding the concept of Riemann sums, limits, and integration, we can calculate the integral of the limit in a Riemann sum for a given function over a specified interval.