Lec 24 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by addressing a previous mistake regarding the terminology of energy and heat, clarifying that in physics, energy, heat, and work are essentially the same concept but measured in different units. The lecture then transitions into an example involving a dartboard to illustrate a probability problem. The scenario involves a target (dartboard), a person (little brother), and a dart thrower, aiming to calculate the likelihood of the person being hit by a dart. The professor introduces the concept of normal distribution, represented by the function e^(-r^2), to model the distribution of dart hits as a function of the radius from the center of the dartboard.

The professor elaborates on the dartboard example by defining the 'part' and the 'whole' in the context of probability. The 'part' refers to the area where the little brother is standing, while the 'whole' encompasses all possible areas where a dart might land. The focus is on calculating the likelihood of darts hitting within specific concentric circles defined by radii r_1 and r_2. The method of shells is introduced to calculate the volume of revolution, which represents the likelihood of hits in a given section. The professor also explains why the height of the volume element is represented by the function e^(-r^2) and the circumference by 2 pi r, multiplied by the thickness dr.

The discussion continues with the interpretation of the dartboard hits' probability distribution. The professor explains that the number of hits in a given area is proportional to the area itself and a constant of proportionality, which depends on the number of attempts made. The concept of the 'mode' is introduced, indicating the most likely spot to be hit (the center), but clarifies that the area around the center is small, making it less likely to be hit when considering the total volume. The professor also addresses the practicality of the chosen function e^(-r^2) by referencing its historical use in modeling the distribution of V-2 rocket hits during World War II.

The professor introduces the 'little brother' into the dartboard model, explaining that the child is standing at a distance approximately twice the radius of the target, away from the dartboard. The brother's location is approximated by a sector of the dartboard's radius. The probability of the brother being hit by a dart is then related to the likelihood of the darts landing outside the target area. The professor emphasizes that the dart thrower often misses the target, which is a significant factor in the probability calculation.

The professor provides a mathematical approach to calculating the probability of hitting the target and, by extension, the little brother. The 'part' of the probability is determined by integrating the function e^(-r^2) between the radii r_1 and r_2, resulting in the expression pi * (e^(-r_1^2) - e^(-r_2^2)). The 'whole' is considered to be the range from 0 to infinity, simplifying the calculation to c * pi * (1 - 0), where c is a constant that cancels out in the final probability ratio. The probability of landing on the target within a radius between r_1 and r_2 is thus the difference in the exponentiated terms, highlighting the idealization used in the model.

The professor concludes the dartboard example by calculating the specific probability of the little brother being hit, given that the target's radius is 'a' and the brother stands between 2a and 3a from the center. The probability is derived to be approximately 1%, based on the model. The lecture then transitions into a new topic: numerical integration. The professor briefly introduces Riemann sums as a method for approximating integrals when analytical solutions are not available, setting the stage for a more in-depth discussion of numerical integration techniques in subsequent lectures.

The professor delves into the topic of numerical integration, starting with a review of Riemann sums, which involve dividing the integration interval into small segments and approximating the area under the curve by summing the products of function values and segment widths. The left and right Riemann sums are described, highlighting their limitations due to inefficiency. The professor then introduces the trapezoidal rule as an improvement over Riemann sums, which involves approximating the area under the curve with trapezoids instead of rectangles. The trapezoidal rule is presented as a more reasonable approach, though it is still not the most efficient method available.

The professor simplifies the formula for the trapezoidal rule, which involves taking the average of the left and right Riemann sums, and presents it as a more efficient method than Riemann sums. However, it is noted that the trapezoidal rule is still not as accurate as desired. The lecture then introduces Simpson's Rule, a more advanced technique for numerical integration that involves fitting parabolas between intervals of the curve. Simpson's Rule is highlighted as a significant improvement over the trapezoidal rule, offering a better approximation for most functions without inflection points.

The professor provides a detailed explanation of Simpson's Rule, which requires the number of intervals to be even and involves calculating the area under a parabola that fits through three points at a time. The formula for the average height under the parabola is given as a weighted average of the function values at those points: (y_0 + 4y_1 + y_2) / 6. The full formula for Simpson's Rule is presented, emphasizing the pattern of weights 1, 4, and 2 that alternates across the intervals. The rule is praised for its effectiveness in approximating integrals, especially when compared to the less efficient Riemann sums and the trapezoidal rule.

The professor concludes the current lecture by summarizing the key points discussed, including the dartboard probability model and the introduction of numerical integration methods. A preview of the next lecture is given, where the professor promises to demonstrate the effectiveness of Simpson's Rule compared to other numerical integration techniques. The lecture ends with an open invitation for any remaining questions from the students.