Problem 16 - Calculus: Early Transcendentals

Question

Use a calculator or computer to make a table of values of left and right Riemann sums \( L_n \) and \( R_n \) for the integral \( \displaystyle \int^2_0 e^{-x^2} \, dx \) with \( n \) = 5, 10, 50, and 100. Between what two numbers must the value of the integral lie? Can you make a similar statement for the integral \( \displaystyle \int^2_{-1} e^{-x^2} \, dx \)? Explain.

Step-by-Step Solution

Answer

For each integral, the value lies between \( L_n \) and \( R_n \) for corresponding \( n \). The range of values tightens with more subintervals.

Step 1: Understanding Riemann Sums

A Riemann sum approximates the value of a definite integral by summing up products of function values and small widths over an interval. Left Riemann sums, \( L_n \), use the left-hand endpoints of subintervals, while right Riemann sums, \( R_n \), use the right-hand endpoints.

Step 2: Divide the Interval

For the integral \( \int_0^2 e^{-x^2} \, dx \), divide the interval \([0, 2]\) into \(n\) subintervals. The width of each subinterval, \( \Delta x \), is \( \frac{2-0}{n} = \frac{2}{n} \).

Step 3: Calculate Left Riemann Sum \( L_n \)

For \( L_n \), sum the areas using left endpoints. The formula is:\[ L_n = \sum_{i=0}^{n-1} e^{-(x_i)^2} \Delta x \]Where \( x_i = 0 + i \times \frac{2}{n} \). Use a calculator to compute \( L_n \) for \( n = 5, 10, 50, 100 \).

Step 4: Calculate Right Riemann Sum \( R_n \)

For \( R_n \), sum the areas using right endpoints. The formula is:\[ R_n = \sum_{i=1}^{n} e^{-(x_i)^2} \Delta x \]Where \( x_i = 0 + i \times \frac{2}{n} \). Compute \( R_n \) for \( n = 5, 10, 50, 100 \) using a calculator.

Step 5: Table of Values

Fill in the computed values of \( L_n \) and \( R_n \) for each \( n \) in a table:\[\begin{array}{|c|c|c|}\hlinen & L_n & R_n \\hline5 & L_5 & R_5 \10 & L_{10} & R_{10} \50 & L_{50} & R_{50} \100 & L_{100} & R_{100} \\hline\end{array}\]

Step 6: Determine Range of Integral Value

The true value of the integral lies between the left and right Riemann sums, i.e., \[L_n \leq \int_0^2 e^{-x^2} \, dx \leq R_n\]For each \( n \), note the interval where the integral value must lie. More subintervals give a tighter estimate.

Step 7: Analyzing Integral from \(-1\) to \(2\)

Repeat Steps 2 to 6 for the integral \( \int_{-1}^{2} e^{-x^2} \, dx \). Note that the range will similarly tighten with increasing \( n \). The same logic of Riemann sums applies to identify the integral value bounds.

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve, between two specific points on the x-axis. In mathematical terms, it's a way to sum the infinitesimal changes over an interval. This action gives us the total accumulation of a quantity, such as area, over that interval.

For example, in the integral \( \int_0^2 e^{-x^2} \, dx \), we are interested in the total area under the curve of \( e^{-x^2} \) from \( x = 0 \) to \( x = 2 \). This area approximates the true value that we seek. Definite integrals are foundational in calculus because they provide exact numerical results for various problems in physics, engineering, and applied mathematics.

When we can't calculate an integral exactly, as in cases involving complex functions like \( e^{-x^2} \), numerical methods such as Riemann sums become invaluable tools for approximation.

Left Riemann Sum

The Left Riemann Sum, \( L_n \), is a method for approximating the value of a definite integral. This technique involves dividing the area under a curve into several rectangles with their heights determined by the left endpoint of each subinterval.

Imagine your interval as being cut into smaller equal parts, or subintervals. If we have an integral from 0 to 2, such as \( \int_0^2 e^{-x^2} \, dx \), and divide it into \( n \) intervals, each small rectangle begins at the left endpoint of its respective subinterval:

• The width of each rectangle is \( \Delta x = \frac{2}{n} \).
• The height is the value of the function at the left endpoint, \( e^{-(x_i)^2} \).

This process can be repeated for various values of \( n \), and as \( n \) increases, \( L_n \) tends to provide a better approximation of the definite integral.

Right Riemann Sum

Right Riemann Sum, \( R_n \), mirrors the Left Riemann Sum concept but uses the right-hand endpoint of each subinterval instead. This method approximates the area under a curve by summing up the areas of rectangles whose heights are defined by the function’s value at the right endpoint.

Considering the same integral \( \int_0^2 e^{-x^2} \, dx \), we divide the interval into \( n \) subintervals:

• The width \( \Delta x = \frac{2}{n} \)
• Heights are taken from \( e^{-(x_i)^2} \), where \( x_i \) is the right endpoint.

Despite the shift in endpoint, as with the Left Riemann Sum, increasing \( n \) results in the \( R_n \) becoming closer to the actual integral value. Together, both \( L_n \) and \( R_n \) create bounds for the true integral, helping to estimate with greater precision.

Numerical Integration

Numerical Integration is the process of approximating the value of a definite integral when an exact solution is challenging or impossible to achieve analytically. Methods like Riemann sums, including the Left and Right approaches, fall under this umbrella of techniques.

This method is critical when working with functions that don't have simple antiderivatives, such as \( e^{-x^2} \). Here are steps involved:

• Divide the integration interval into smaller segments or subintervals.
• Apply numerical methods, such as using left or right rectangle approximations.
• Calculate the approximations for these subintervals and sum them to obtain an approximate integral value.

As you increase the number of divisions, or \( n \), the approximation improves. This approach is widely applicable in scientific computations, where exact integration isn't feasible, providing a practical solution by narrowing the bounds within which the integral value is found.