Lec 21 | MIT 18.01 Single Variable Calculus, Fall 2007

The script begins with an introduction to MIT OpenCourseWare, emphasizing its commitment to providing free, high-quality educational resources. Viewers are encouraged to support and visit the platform at ocw.mit.edu. The professor then corrects a previous error regarding the calculation of the average value, highlighting the importance of including a factor of 'n' during integration for a Riemann sum over the interval from 0 to n. This correction is crucial as it impacts exam performance, and the professor intends to delve deeper into the concept of average value in future lectures.

The script continues with a detailed discussion of the Fundamental Theorem of Calculus, specifically its second version, which states that the derivative of an integral can restore the original function. The professor uses this theorem to solve a differential equation, y' = 1/x, by integrating and defining the integral as the logarithm function. This approach allows for the derivation of logarithmic properties, such as its derivative being 1/x and its value at specific points. The professor also explains how to graph the logarithm function using its derivative and a single value, emphasizing the function's increasing nature and its behavior as x approaches 0 and infinity.

The script delves into the properties of the logarithm function, focusing on its graph and behavior. The professor discusses the function's concavity, its increasing nature, and its limits as x approaches 0 and infinity. The concept of 'e' is introduced as the value for which the logarithm function equals 1. The professor also explains why the logarithm function is negative for x < 1, using both the function's increasing nature and an integral manipulation approach. The script concludes with an advanced property of logarithms, demonstrating how integral properties can be manipulated to show that L(ab) equals the sum of L(a) and L(b).

The script introduces more complex functions that cannot be expressed in terms of known elementary functions. The focus is on understanding these functions by examining their properties, such as derivatives and values at specific points. The professor discusses the function F(x), which is defined as an integral involving an exponential function. The function's increasing nature, concavity, and odd symmetry are highlighted. The script also introduces the error function (erf), which is related to the integral of an exponential function and is associated with the normal distribution. The professor emphasizes the importance of understanding the behavior of these functions at infinity and their significance in various mathematical fields.

The script shifts focus to the role of integrals in calculating areas between curves, which is more closely associated with the first Fundamental Theorem of Calculus. The professor explains how to set up integrals to find areas by considering the curves as the boundaries of the region of interest. The method involves identifying the integrand, the limits of integration, and the bounds of the region. The script emphasizes the importance of these elements in calculating the area accurately and provides an example of finding the area between two curves, x = y^2 and y = x - 2, by setting up the appropriate integral.

The script discusses a method for calculating the area between two curves by using vertical slices. The professor illustrates this with the curves x = y^2 and y = x - 2, explaining the need to split the area into two parts due to the different formulas for the lower function. The intersection points of the curves are identified, and the formulas for the top and bottom functions in each region are established. The area calculation is then set up as two separate integrals, one for each half of the area between the curves. The professor emphasizes the importance of distinguishing between the top and bottom functions and the correct application of the limits of integration.

The script presents an alternative method for calculating the area between curves using horizontal slices. This method involves reversing the roles of x and y, writing x as a function of y, and setting up the integral in terms of dy. The professor demonstrates how to find the area between the same curves, x = y^2 and y = x - 2, by identifying the limits of y and the difference between the rightmost and leftmost points. The integral is then set up from y = -1 to y = 2, and the difference between the right and left functions is used as the integrand. The professor shows that this method leads to a simpler integral and concludes the script with the final calculation, yielding the area as 9/2.