Lec 25 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by acknowledging the support of the audience, which enables MIT OpenCourseWare to provide high-quality educational resources for free. Donations and additional materials can be found at ocw.mit.edu. The lecture then delves into numerical methods, specifically numerical integration, using the integral from 1 to 2 of dx/x as a simple example. The exact solution, ln(2), is contrasted with the numerical approximation obtained from a calculator, approximately 0.693147. The discussion highlights the accuracy achievable through numerical integration methods and the simplicity of the function 1/x in this context.

The professor compares two numerical integration methods: the trapezoidal rule and Simpson's Rule. The trapezoidal rule is demonstrated with a simple two-interval example, resulting in an approximation of 0.96, which is notably less accurate than the true value. Simpson's Rule is then introduced, claiming it to be more accurate, especially in this case, yielding an approximation of 0.69444, very close to the actual value of ln(2). The professor explains that Simpson's Rule works by matching a parabola to the curve, providing a more accurate estimation due to its higher-order validity. However, a caution is given regarding the limitations of Simpson's Rule when dealing with functions that have singularities or are not smooth, particularly near x equals 0.

The professor elaborates on Simpson's Rule's accuracy, noting that the error is approximately of the size of (delta x)^4, implying that increasing the number of intervals significantly improves accuracy. The rule's effectiveness is attributed to its derivation, which ensures exact answers for all parabolas and, coincidentally, for cubics as well. The professor emphasizes the importance of function smoothness and bounded derivatives for the rule to work effectively. A mnemonic device is suggested to remember the weights used in Simpson's Rule by checking it against the simplest case where the function f(x) equals 1.

The lecture shifts focus to the calculation of a mysterious constant, the square root of pi / 2, by using slicing methods. The professor reminds students of a previous computation involving the volume under the bell-shaped curve e^(-r^2) when rotated around an axis. The new problem involves calculating an area, which is approached by reimagining the volume calculation using slices. The professor hints at a trick where the volume V is found to be equal to the square of the area Q, leading to the conclusion that Q equals the square root of pi, once the volume V is known.

The professor connects the concept of asymptotic behavior to the slicing method used for calculating volumes and areas. The function discussed has an asymptote approaching the square root of pi / 2 as x goes to infinity. The professor illustrates the connection by showing that the area Q is twice the value of the function at infinity, thus confirming the earlier claim that the area under the curve from 0 to infinity is the square root of pi. The visualization of the function's graph and its asymptotic behavior is used to clarify the relationship between the volume and the area.

The professor introduces a three-dimensional perspective to visualize the slicing method for calculating volume. The x, y, and z axes are defined, with the z-axis representing the height of the slices formed by the rotation of the function e^(-r^2) around the z-axis. The professor explains the concept of a slice A(b) and how it relates to the volume calculation. The formula for the volume by slices is presented as the sum of the areas of the slices multiplied by their respective thicknesses, extending from negative infinity to positive infinity.

The professor focuses on calculating the area of a slice, A(b), by considering the function e^(-r^2) and expressing it in terms of b and x. Using the right triangle relationship, the professor derives the formula for the height of the slice and applies the law of exponents to separate the variables. The area A(b) is then found to be a constant times the integral of e^(-x^2) from negative infinity to positive infinity, which is a known quantity. The total volume is expressed as the integral of A(b) over all b's, leading to the conclusion that the volume is proportional to the square of the area Q.

The professor addresses a student's question regarding the independence of variables x and y in the context of slicing methods. The professor explains that while x and y are symmetric and can be treated equally, the choice to fix y and vary x separates their roles. The professor clarifies that the constant Q, derived from the area A(b), does not depend on y and that x varies independently. The discussion emphasizes the flexibility of choosing the slicing method and the importance of understanding the roles of variables in these calculations.

The professor provides a review of the exam content, outlining the five questions that students can expect. The exam will cover the calculation of definite integrals using the fundamental theorem of calculus, numerical approximations through Riemann sums, trapezoidal rule, and Simpson's rule, as well as the calculation of areas, volumes, and other cumulative sums. The professor also discusses the difference between upper and lower sums, and right and left sums, using the function y = 1/x as an example. The explanation aims to clarify these concepts for the students before the exam.

The professor addresses the complexities of determining asymptotic behavior and choosing the appropriate method for calculating areas and volumes. The professor assures students that the decision on which numerical approximation method to use will be clear on the exam. For areas and volumes, students are expected to work back to a two-dimensional diagram and decide whether to integrate with respect to dx or dy. The professor uses an example involving the curve y between 0 and x - x^3 to illustrate the process of elimination when choosing between different integration methods, emphasizing the importance of selecting the simplest and most feasible method.