Question

Numerical Reasoning Consider a trapezoid of area 4 bounded by the graphs of \(y=x, y=0, x=1,\) and \(x=3\) (a) Sketch the region. (b) Divide the interval [1,3] into \(n\) subintervals of equal width and show that the endpoints are \(1<1+1\left(\frac{2}{n}\right)<\cdots<1+(n-1)\left(\frac{2}{n}\right)<1+n\left(\frac{2}{n}\right)\) (c) Show that \(s(n)=\sum_{i=1}^{n}\left[1+(i-1)\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)\) (d) \(S(n)=\sum_{i=1}^{n}\left[1+i\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)\) that Show (e) Complete the table. \begin{tabular}{|l|l|l|l|l|} \hline\(n\) & 5 & 10 & 50 & 100 \\ \hline\(s(n)\) & & & & \\ \hline\(S(n)\) & & & & \\ \hline \end{tabular} (f) Show that \(\lim _{n \rightarrow \infty} s(n)=\lim _{n \rightarrow \infty} S(n)=4\)

Short answer

Both sequences \( s(n) \) and \( S(n) \) converge to 4, which is the area of the trapezoid as defined by the lines \( y=x, y=0, x=1, x=3 \). This means that as the number of subintervals increases (i.e., as \( n \) goes to infinity), the approximate area under the curve \( y = x \) between \( x = 1 \) and \( x = 3 \) (represented by smaller trapezoids) also converges to the true area of the trapezoid (4).

Step-by-step solution

Sketch the Region

The graphs of \( y = x \), \( y = 0 \), \( x = 1 \), and \( x = 3 \) form a trapezoid. On the x-axis are the lines \( x = 1 \) and \( x = 3 \), and the upper sides are given by \( y = x \), which meet at (1,1) and (3,3).

Divide the Interval

The interval [1,3] can be divided into n equal subintervals, with each endpoint given by \( 1+i\left(\frac{2}{n}\right) \), where i=0,1,2,3,...,n. The difference between each endpoint is \( \frac{2}{n} \), resulting in n equal subintervals.

Compute the Sums s(n) and S(n)

To compute \( s(n) \) and \( S(n) \), it's important to remember that each sum is the area under the curve \( y = x \) for a specific subinterval. For \( s(n) \), it's the area under the curve up to the left endpoint of each subinterval, while for \( S(n) \), it's the area under the curve up to the right endpoint of each subinterval. The mathematical expressions to calculate them are given by \(s(n)=\sum_{i=1}^{n}\left[1+(i-1)\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)\) and \(S(n)=\sum_{i=1}^{n}\left[1+i\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)\) respectively.

Complete the Table

Next, use the formulas from Step 3 to complete the table for \( s(n) \) and \( S(n) \) when \( n \) is 5, 10, 50, and 100. Note that as \( n \) increases, you will start to see that both \( s(n) \) and \( S(n) \) are getting close to 4.

Show Limits

Finally, show that \( s(n) \) and \( S(n) \) both approach 4 as \( n \) approaches infinity. This aligns with the actual area of the trapezoid. This can be proven by substituting into the formulas for \( s(n) \) and \( S(n) \), simplifying and taking the limit as \( n \) approaches infinity.

Key concepts

Limit of a Sequence

Understanding the limit of a sequence is fundamental in calculus. It refers to the value that a sequence's terms get closer to as the number of terms increases indefinitely. Imagine you're taking steps towards a wall; as you take smaller and smaller steps, you get closer to the wall without actually touching it. Mathematically, if you have a sequence \(a_n\), and there's a number L such that \(a_n\) gets as close as we want to L by taking enough terms of the sequence, we say that the limit of \(a_n\) as n approaches infinity is L, denoted as \(\lim_{n \to \infty} a_n = L\).

In the exercise, the notion of limit is used to confirm that the sum of the areas of trapezoids with increasing numbers of intervals (\(n\)) approaches the actual area under the curve as \(n\) approaches infinity. This ties into the idea that, theoretically, if we could create infinite trapezoids with infinitesimally small width, we'd perfectly match the area under the curve.

Sum of a Series

The concept of the sum of a series is closely related to the limit of a sequence. A series is the summation of the terms of a sequence. When this summation is infinite, we often want to find out if there's a finite value that the sum approaches, which is known as the series' limit. Mathematically, a series is usually denoted as \(S = \sum_{n=1}^{\infty} a_n\), with \(a_n\) being the nth term of the sequence.

In numerical integration, like in our exercise, we are summing up the areas of several small shapes (like rectangles, trapezoids, etc.) to approximate the area under a curve. Both \(s(n)\) and \(S(n)\) are examples of finite series representing the sum of the areas of trapezoids formed between the curve and the x-axis, using the left and right endpoints of subintervals respectively.

Area Under a Curve

The area under a curve is a central concept in calculus and numerical integration. It represents the integral of a function over a certain interval. In a graphical sense, if you shade the region between the curve and the x-axis, you're shading the area under the curve. Such areas can represent various things in real-world applications, like distance traveled over time as given by the speed function.

The trapezoid in our exercise is a geometric representation of the area under the straight line \(y = x\) between \(x = 1\) and \(x = 3\). Numerical integration methods, such as Riemann sums, aim to find the area under a curve when we cannot find the exact integral, or when we are dealing with data points rather than a continuous function.

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve. This technique involves dividing the region into shapes (rectangles or trapezoids), computing their individual areas, and then summing those areas. The result is an approximation because these shapes do not fit perfectly under the curve, but by increasing the number of divisions (as \(n\) increases), the fit becomes better.

In the exercise, both \(s(n)\) and \(S(n)\) are specific instances of Riemann sums using trapezoids. They approximate the area under the curve \(y = x\) from x = 1 to x = 3. As we increase \(n\), the width of each trapezoid gets smaller, resulting in a more accurate approximation of the area. The exercise guides the student to see that by letting \(n\) approach infinity, the sum of these areas will converge to the actual area of the trapezoid, reinforcing the idea that Riemann sums become more precise with finer divisions.