Numerical Reasoning Consider a triangle of area 2 bounded by...

Numerical Reasoning Consider a triangle of area 2 bounded by the graphs of $y=x, y=0,$ and $x=2.$ (a) Sketch the region. (b) Divide the interval [0,2] into $n$ subintervals of equal width and show that the endpoints are $0<1\left(\frac{2}{n}\right)<\dots<(n-1)\left(\frac{2}{n}\right)< n\left(\frac{2}{n}\right).$ (c) Show that $s(n)=\sum_{i=1}^{n}\left[(i-1)\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right).$ (d) Show that $S(n)=\sum_{i=1}^{n}\left[i\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right).$ (e) Complete the table. $$\begin{array}{|l|c|c|c|c|}\hline n & 5 & 10 & 50 & 100 \\\hline s(n) & & & & \\\hline S(n) & & & & \\\hline\end{array}$$ (f) Show that $\lim _{n \rightarrow \infty} s(n)=\lim _{n \rightarrow \infty} S(n)=2$.

Numerical Reasoning Consider a triangle of area 2 bounded by the graphs of $y=x, y=0,$ and $x=2.$
(a) Sketch the region.
(b) Divide the interval [0,2] into $n$ subintervals of equal width and show that the endpoints are $0<1\left(\frac{2}{n}\right)<\dots<(n-1)\left(\frac{2}{n}\right)< n\left(\frac{2}{n}\right).$
(c) Show that $s(n)=\sum_{i=1}^{n}\left[(i-1)\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right).$
(d) Show that $S(n)=\sum_{i=1}^{n}\left[i\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right).$
(e) Complete the table.
$$\begin{array}{|l|c|c|c|c|}\hline n & 5 & 10 & 50 & 100 \\\hline s(n) & & & & \\\hline S(n) & & & & \\\hline\end{array}$$
(f) Show that $\lim _{n \rightarrow \infty} s(n)=\lim _{n \rightarrow \infty} S(n)=2$.

Key Concepts

Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over an interval by dividing the region into smaller subregions (usually rectangles), calculating the area of each piece, and then summing these areas. This concept is fundamental in understanding how integrals can be approximated numerically and how, in the limit as the number of subdivisions increases, the sum approaches the exact area under the curve.

Partitioning an Interval

Partitioning an interval involves dividing the interval into a number of equal or unequal subintervals. In numerical integration, choosing these subintervals carefully (often of equal width) is crucial, as it lays the groundwork for constructing Riemann sums. The endpoints of these subintervals are then used to evaluate the function and approximate the area beneath its graph.

Summation Notation and Series

Summation notation is a concise way to represent the addition of a sequence of terms. In the context of numerical integration, it is used to express the total sum of the areas of the subregions, whether the heights of these regions are taken at the left endpoints, right endpoints, or other points in the subintervals. This formalism is key to transitioning from a finite sum to the concept of an integral.

Limits and Convergence of Sums

Taking the limit of a Riemann sum as the number of subintervals approaches infinity is a critical step in numerical integration. This process demonstrates that as the partitions become finer, the Riemann sum converges to the exact value of the definite integral. Understanding this limit is fundamental in calculus, as it bridges finite approximations with the continuous nature of integrals.

Graphical Representation of Regions

Graphically representing the region of integration, such as sketching the area bounded by given curves, is an important concept in understanding the geometry and bounds of an integral. This visualization aids in setting up the appropriate limits of integration and provides context for how the numerical approximations (via Riemann sums) relate to the actual area under the curve.

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