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Riemann sums are a method for approximating the definite integral (area under a curve) of a function, f(x), over a specified interval [a, b]. To perform a Riemann sum, the interval [a, b] is divided into a number of smaller subintervals, and the area of a rectangle is calculated for each subinterval. The height of each rectangle is determined by the function's value at a specific point within the subinterval. The sum of these rectangular areas provides an approximation of the area under the curve. There are various types of Riemann sums, such as left, right, and midpoint, depending on which point within the subinterval is used to determine the height.

As the number of subintervals increases, the width of each rectangle decreases. This in turn leads to the rectangles more closely following the shape of the curve, providing a better approximation of the area under the curve. In other words, increasing the number of subintervals causes the Riemann sum to approach the true value of the definite integral, and therefore the approximation error decreases.

To illustrate this concept, consider a function f(x) for which we want to approximate the area under the curve over the interval [a, b]. Let's start with a Riemann sum using n=4 subintervals. The approximation will be relatively coarse, and there will be noticeable gaps or overlaps between the rectangles and the curve, leading to a larger error in the approximation. Now, let's increase the number of subintervals to n=8. The width of each rectangle will be smaller, allowing them to better follow the curve and resulting in a more accurate approximation of the area under the curve. In general, as the number of subintervals approaches infinity (n->∞), the Riemann sum approaches the true value of the definite integral, which can be expressed as: \[ \int_a^b f(x) dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i^*) \Delta x \] Where the sum represents the sum of the rectangular areas, and f(x_i^*) represents the height of the rectangle in the i-th subinterval.

In conclusion, Riemann sum approximations to the area of a region under a curve improve as the number of subintervals increases, because the rectangles more closely follow the shape of the curve. As the number of subintervals approaches infinity, the approximation error decreases, and the Riemann sum converges to the true value of the definite integral.