Lec 19 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by introducing the Fundamental Theorem of Calculus (FTC), emphasizing its importance in calculus. The FTC connects differentiation and integration, stating that if a function F is the antiderivative of f(x), then the definite integral from a to b of f(x) is F(b) - F(a). The professor explains two notations for this theorem and provides an example using the function F(x) = x^3/3 to illustrate how the FTC simplifies the calculation of integrals.

This section continues with practical examples to demonstrate the FTC's utility. The professor calculates the area under a curve for a function like sin(x) from 0 to π, highlighting the ease with which the FTC can handle problems that would be complex with Riemann sums. The discussion also includes the integral of x^100 from 0 to 1, showcasing the theorem's versatility. The professor then offers an intuitive interpretation of the FTC relating to distance traveled, using the concepts of position and velocity.

The professor provides an intuitive interpretation of the FTC using the analogy of distance and speed. By considering x(t) as position and x'(t) as speed, the FTC is shown to equate the integral of velocity over a time interval to the change in position. The connection to Riemann sums is also discussed, with a thought experiment illustrating how checking speed at one-second intervals can approximate the total distance traveled, aligning with the FTC's principles.

The professor extends the concept of integration to include negative values, emphasizing that the formulas remain the same but the interpretation changes. The definite integral is redefined to account for areas above and below the x-axis, with a clear example using the sine function from 0 to 2π. The integral's result is zero, demonstrating how positive and negative areas cancel each other out, reflecting no net change in position.

The focus shifts to the properties of integrals, starting with the additive property, which allows the integration of a sum of functions to be broken into separate integrals. The professor also discusses the historical reasoning behind the integral symbol's design, relating it to the sum symbol. Constant factors can be pulled out of integrals, and the properties are established to maintain consistency with the FTC.

The professor explains additional properties of integrals, such as the ability to integrate over intervals in any order due to a new definition, and the fact that the integral from a to a is zero. The fifth property introduces the concept of integrals in reverse order, defining it as the negative of the integral in the forward direction. These properties are crucial for maintaining consistency within calculus operations.

The sixth property, estimation, is introduced, which allows for comparing the integrals of two functions where one is less than or equal to the other. The professor illustrates this with an example, integrating the inequality e^x >= 1 from 0 to b, resulting in e^b >= 1 + b. The importance of the direction of integration (a < b) is highlighted, and the potential for error when dealing with negative values is noted.

The final topic is the change of variables in integration, also known as substitution. The professor outlines the process and emphasizes the importance of maintaining the correct limits when substituting variables. A common mistake is pointed out, where ignoring the sign change in the derivative can lead to incorrect results. The professor advises breaking the integral into pieces where the derivative maintains a consistent sign to avoid confusion.