Lec 18 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by introducing the third unit on integration, focusing on definite integrals as the second half of calculus following differentiation. Definite integrals are presented from a geometric perspective, aiming to calculate the area under a curve between two points, a and b. The concept is explained with the notation involving limits and the function f(x), represented as ∫_a^b f(x) dx. The idea is to approximate this area by dividing it into rectangles and taking the limit as these rectangles become infinitesimally thin, thus getting closer to the actual area under the curve.

The paragraph delves deeper into the geometric interpretation of definite integrals, introducing the Riemann sum as a method to calculate areas under a curve. The process involves dividing the area into rectangles and summing their areas, which is simplified by taking the limit as the width of these rectangles approaches zero. The professor illustrates this with a graph, dividing the x-axis into increments and using a staircase pattern to form rectangles. The base of each rectangle is calculated as b/n, where n is the number of divisions, and the height is determined by the function f(x) at each point, leading to a summation of areas that approximates the integral.

The professor continues with the calculation of definite integrals, using the function f(x) = x^2 as an example. The process involves setting up a summation of the areas of rectangles formed under the curve. The base of the rectangles is b/n, and the height is given by the function evaluated at different points. The summation is then simplified by factoring out common terms and using the limit as n approaches infinity to find the exact area under the curve. The professor also introduces summation notation to compress the formula and make it more manageable.

In this paragraph, the professor discusses a geometric trick to estimate the summation of squares, which is part of the integral calculation. By visualizing the summation as a pyramid with a base of n by n blocks and height n, the professor shows how to estimate the size of the summation. The trick involves comparing the staircase pattern to a solid pyramid and using the formula for the volume of a cone to find that the summation is greater than 1/3 n^3 and less than 1/3 (n+1)^3. This estimation helps in taking the limit as n approaches infinity to find the exact value of the integral.

The professor provides a detailed analysis of the summation of squares and its relation to the integral of x^2 from 0 to b. By simplifying the expression and taking the limit as n approaches infinity, the professor demonstrates that the integral is equal to 1/3 b^3. The explanation includes a discussion on why certain terms are factored out and how the limit helps in simplifying the complex summation to a more manageable form. The importance of recognizing patterns in summations and their relation to integrals is emphasized.

The paragraph introduces summation notation and its application in expressing and simplifying integral calculations. The professor shows how to use the summation notation to represent the sum of i^2 from i=1 to n, which tends to 1/3 as n approaches infinity. The professor also discusses the pattern observed in the results of integrals for functions like x^0, x^1, and x^2, suggesting that the integral of x^n might be b^(n+1)/(n+1). This insight is used to predict the result for the integral of x^3.

The professor reflects on the historical context of calculating areas under curves, mentioning Archimedes' complex method for finding the area under a parabola. The professor suggests that the brilliance of Archimedes' method may have inadvertently delayed the development of simpler calculus techniques for 2000 years. The paragraph highlights the power of calculus in providing easy methods for calculating areas, which were previously a laborious process.

The professor introduces Riemann sums as a general procedure for calculating definite integrals. The process involves dividing the interval [a, b] into equal parts and summing the areas of rectangles formed by choosing a height f(c_i) within each interval. The notation is explained, with delta x representing the width of each rectangle, which approaches zero as the number of rectangles increases. The Riemann sum is shown to be a precursor to the integral notation used by Leibniz, with dx replacing delta x in the limit.

The professor provides a financial example to illustrate the concept of integrals as cumulative sums. The example involves borrowing money at a varying rate over a year, with the total amount borrowed represented as a sum of daily borrowings. The sum is analogous to the integral from 0 to 1 of f(t) dt, where f(t) represents the borrowing rate per year. The professor also discusses how the amount owed at the end of the year, with continuous compounding, can be modeled using an integral.

In the final paragraph, the professor emphasizes the importance of integrals in financial modeling, particularly in strategizing and optimizing borrowing, spending, and investing. The example given involves calculating the total amount owed at the end of the year with continuous compounding interest. The integral from 0 to 1 of e^(r(1-t)) f(t) dt is introduced to represent this calculation, highlighting the relevance of integrals in financial decision-making.