BusCalc 13.4 Riemann Sums

The paragraph begins with an introduction to solving integration problems using the method of integration by parts, a technique previously discussed. The speaker simplifies the problem by taking the constant factor 'four' out of the integrand, which is permissible when dealing with derivatives or anti-derivatives. A table is constructed to apply the integration by parts formula, with the left column representing the function of x cubed and the right column representing the exponential function e to the 2x. The derivatives and anti-derivatives of these functions are calculated, leading to a final expression for the anti-derivative of the given function, which includes a constant of integration 'c'. The paragraph concludes with an invitation for questions, transitioning into the next problem involving finding the anti-derivative of a different function.

The second paragraph focuses on applying integration by parts to a function involving a natural logarithm. The speaker outlines the process of setting up the integration by parts formula with 'natural log of t' and 't squared' as the initial functions. The derivatives and anti-derivatives are calculated, leading to a step involving the anti-derivative of a horizontal grouping. The process concludes with the addition of a constant 'c'. It is highlighted that when dealing with natural logs in integration by parts, only two rows are needed in the table, and the natural log should always be placed in the first column to avoid needing to find the anti-derivative of the natural log itself. The paragraph ends with a teaser for the next topic: Riemann sums.

The third paragraph introduces Riemann sums as a method for calculating the areas of irregular shapes that cannot be broken down into simple geometric shapes like triangles or rectangles. The process involves dividing the area under the curve into 'n' rectangles and using the function's value at the left side of each rectangle as the height. The area of each rectangle is calculated and then summed to estimate the total area under the curve. The concept of relative error is discussed, showing how it decreases as the number of rectangles increases, providing a more accurate estimate. The actual area under the curve is revealed to be 5.39, and the process is demonstrated with a step-by-step calculation using three and then eight rectangles to illustrate the decreasing error.

The fourth paragraph delves deeper into the concept of using more rectangles to decrease the relative error and approximate the area under a curve more closely. It is shown that with 20 rectangles, the error is about two percent, and with 30 rectangles, it drops to 2.5 percent. The speaker uses Desmos to create graphics for visual aid and explains how the choice of the function and the number of rectangles can affect the error. The paragraph concludes with a table summarizing that as the number of rectangles increases, the relative error approaches zero, and the estimate using the areas of the rectangles approaches the actual area of the shape.

The fifth paragraph explains the algebraic process of calculating the width of each rectangle used in the Riemann sum and how to express the height of each rectangle in terms of the function evaluated at specific points. The width of each rectangle is determined by dividing the interval length (b - a) by the number of rectangles (n). The height of each rectangle is the function evaluated at points a + (j - 1) * (b - a) / n, where j is the rectangle number. The paragraph introduces summation notation and explains how to sum the areas of all rectangles to approximate the area under the curve. It concludes with a discussion of how different notations can be used to represent the same mathematical concept.

The sixth paragraph continues the discussion on Riemann sums and introduces definite integrals as the exact area under a curve, which is found by taking the limit as the number of rectangles (n) approaches infinity and the width of each rectangle (delta x) approaches zero. The paragraph explains that as the number of rectangles increases, the error decreases, and the approximation gets closer to the actual area. It also touches on the use of different notations, such as the integral symbol with limits, to represent the process of finding the area under a curve. The concept of a microscopic change in the x variable (dx) is introduced, which is used when the change is so small that it's not observable, akin to the deltas used in derivatives.

The seventh paragraph discusses how calculators use the concept of Riemann sums to perform numerical integration and find the area under a curve. The speaker demonstrates how to use a calculator to find the area under a parabola defined by the function half x squared plus one, between the bounds of zero and one. The calculator's process is described as incrementally increasing the number of rectangles used in the Riemann sum until the change between successive approximations is minimal, indicating a sufficiently accurate result. The paragraph emphasizes that the calculator's numerical integration feature uses an algorithm that adds up the areas of many small rectangles to approximate the integral.

The eighth paragraph illustrates the process of estimating the area under a curve using different numbers of rectangles. The focus is on a parabola represented by the function x squared plus one, with the interval from 0 to 3. The speaker shows how to calculate the area using two, four, and eight rectangles, providing detailed steps for each case. The heights of the rectangles are determined by evaluating the function at specific points, and the areas are calculated by multiplying the heights by the width of the rectangles. The results from using different numbers of rectangles are compared, showing how the approximation improves with more rectangles.

The ninth paragraph continues the process of refining the area approximation under a curve by increasing the number of rectangles to eight. The speaker calculates the height of each rectangle by evaluating the function at the left side of each rectangle and then multiplies the sum of these heights by the width of the rectangles to find the area. The process is shown to be iterative, with the speaker using a calculator to find the necessary function evaluations. The paragraph emphasizes the importance of order of operations and the use of parentheses in calculations, especially when using a calculator.

The final paragraph summarizes the process of calculating areas under a curve using Riemann sums and highlights the importance of understanding the underlying method, even though it may be tedious. The speaker relates the process to the functioning of calculators and computers, which perform similar calculations more efficiently. The paragraph concludes with a note on the use of numerical integration on calculators to find areas under curves and an encouragement for students to practice the method before an upcoming exam.