Lec 33 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by introducing the topic of polar coordinates, focusing on the concept of area in polar coordinates. Using the analogy of a pie, the professor explains how to calculate the area of a slice (delta A) with a radius 'a' and an arc length delta theta. The formula for the area of the pie is given as pi * a^2, and the area of the slice is derived as 1/2 * a^2 * delta theta. The lecture then transitions into discussing the area of a 'variable pie' where the radius r is a function of the angle theta (r = r(theta)), and the method of subdividing the pie into small chunks to calculate the total area is introduced.

The professor delves deeper into the concept of a variable radius in polar coordinates, using the example of a pie with a 'wavy crust'. The idea is to break down the area into small, manageable slices, each with a circular arc. The approximation for the area of each slice, delta A, is given as approximately 1/2 * r^2 * delta theta, with the exact formula being derived from the limit as the slices become infinitesimally small. The total area is then expressed as the integral from a starting angle to an ending angle of 1/2 * r^2 d theta. The professor emphasizes that this formula is particularly useful when r is a function of theta.

An example is presented to illustrate the calculation of the area for a circle in polar coordinates, using the equation r = 2a * cos(theta). The professor explains that the area is only swept out between -pi/2 and pi/2 due to the cosine function being positive in this range and becoming zero at the endpoints. The integral to calculate the area is set up and solved, resulting in pi * a^2, which is the expected area of a circle. The professor also addresses questions from students regarding the determination of the range for integration and the visualization of the polar curve.

The professor discusses what happens outside the range of theta for polar coordinates plotting, specifically beyond pi/2. Using the elbow analogy, the professor explains that as theta increases beyond pi/2, the radius r becomes negative, effectively causing the plot to go backward. It is emphasized that the plot will sweep out the same region twice, which could lead to double counting of the area if the integration range is extended to 2pi or beyond. The professor also addresses the possibility of integrating from 0 to pi, which surprisingly yields the correct area but through a less conventional approach.

The professor provides a detailed walkthrough of plotting the polar function r = sin(2theta), which results in a shape known as a four-leaf rose. By plotting key points at theta = 0, pi/4, and pi/2, the professor demonstrates how the function behaves symmetrically in all quadrants. The importance of understanding the order of plotting is highlighted to grasp the overall shape and behavior of the polar function.

The professor introduces a more complex polar function, r = 1 / (1 + 2cos(theta)), and begins by plotting several points to understand its behavior. Key points are identified at theta = 0, pi/2, and -pi/2, revealing that the function has a denominator of zero at certain angles, indicating asymptotes. The professor then transitions into discussing the conversion of polar coordinates to rectangular coordinates, which is essential for further analysis of the function's shape and characteristics.

The professor demonstrates the conversion of the polar function r = 1 / (1 + 2cos(theta)) into its rectangular coordinate form, resulting in an equation that represents a hyperbola. The connection between the polar formula for area and Kepler's Law in astronomy is highlighted, showing how the rate of change of area swept out by a comet's trajectory around the sun is constant. This leads to a discussion on the conservation of angular momentum and its physical implications, such as the effect on spinning objects.

The professor outlines the topics that will be covered in the upcoming exam, emphasizing three main integration techniques: trig substitution, integration by parts, and partial fractions. The second half of the exam will focus on parametric curves, arc length, area of surfaces of revolution, and polar coordinates, including area calculations. The professor provides guidance on exam preparation, advising students on which techniques to apply and when to seek assistance or avoid particular problems.

The professor discusses advanced topics in integration, specifically focusing on partial fractions and how to handle them when the denominator contains repeated factors or higher-degree terms. The importance of setting up the integral correctly and knowing when to use long division or other algebraic methods is emphasized. The professor also addresses the potential complexity of integration by parts and the use of reduction formulas, advising students on how to approach these techniques during the exam.

The professor provides an example of using integration by parts for a complex integral involving a function that simplifies upon differentiation. The steps of the process are outlined, including the correct identification of u and v', and the subsequent steps to solve the integral. The professor also introduces a shortcut for handling cases where the numerator and denominator are closely related, demonstrating how to simplify the integral and arrive at the final solution.