AP Calculus AB: Lesson 6.2 Riemann and Trapezoidal Sums

Michelle Kremmel introduces the concept of using Riemann and trapezoidal sums to approximate the area under a curve between the x-axis and the curve itself over a specified interval. She explains that instead of counting squares, rectangles of a systematic width are created and their areas are summed up to estimate the total area. The process involves ignoring parts of the graph outside the interval and focusing on the specific region between x=1 and x=7. The method begins with rectangles of width two, and the height of each rectangle is determined by the function value at the left edge of the rectangle where it intersects the curve. The areas of the rectangles are then computed and summed to give an approximation of the total area, which in this case is 42.667, representing a rough estimate rather than the exact area.

The script delves into the specifics of the left Riemann sum method, where the left endpoint of each subinterval is used to determine the height of the rectangles. It acknowledges the imperfect nature of this approximation, as some rectangles may extend outside the curve, leading to an overestimation in some areas and an underestimation in others. The process is repeated with narrower rectangles (width of one) to improve the approximation, resulting in an estimated area of 41.667. The script also discusses the impact of reducing the width of the rectangles, noting that it leads to a more accurate approximation. Additionally, it explains that a left Riemann sum tends to underestimate the area when the function is increasing and overestimate when the function is decreasing.

The video script shifts focus to the right Riemann sum, where the right endpoint of each subinterval is used to determine the height of the rectangles. This method results in a different approximation, with an initial estimate of 34.667, which is visually identified as an underestimate. The script explains that using narrower rectangles with a width of one leads to a more accurate approximation of 37.667. It also discusses the trade-off between the accuracy of the approximation and the complexity of the calculation, especially as the number of rectangles increases. The right Riemann sum is shown to underestimate when the function is decreasing and overestimate when the function is increasing, with the actual area lying between the estimates provided by the left and right Riemann sums.

The midpoint Riemann sum is introduced as a technique that uses the midpoint of each subinterval to determine the height of the rectangles. This method is shown to balance the surplus and deficit areas of the rectangles, providing a more accurate approximation of the area under the curve. The script illustrates this with rectangles of width two, resulting in an estimated area of 40.667. It emphasizes that making the width of the rectangles smaller improves the approximation, as demonstrated with widths of one and 0.5, yielding estimates of 40.167 and 40.042, respectively. The midpoint Riemann sum is noted for its balanced approach, although it's not as clear whether the approximation is an overestimate or an underestimate as it is with the left or right Riemann sums.

The script introduces the trapezoidal sum as a new technique for approximating the area under a curve. Unlike the Riemann sums that use rectangles, the trapezoidal sum uses trapezoids formed by connecting the left and right endpoints of each subinterval to the curve. The area of each trapezoid is calculated as the average of the two bases (function values at the endpoints) times the height (width of the subinterval). The script demonstrates this method with subintervals of width two, resulting in an estimated area of 38.667. It notes that each trapezoid underestimates the actual area, regardless of whether the function is increasing or decreasing, and attributes this to the concavity of the function. The trapezoidal sum is shown to provide a better approximation when using narrower subintervals, with an estimate of 39.667 for width 0.5.

The video script provides practical examples of using the different approximation techniques with a given function, f(x), and its table values. It demonstrates how to apply the left Riemann sum, right Riemann sum, midpoint Riemann sum, and trapezoidal sum to approximate the area under the curve on the interval from 0 to 5. The script emphasizes the importance of understanding the function's behavior to determine whether the approximation is an overestimate or an underestimate. It also highlights the visual aspect of using a graph to verify the results and the algebraic process of calculating the areas of the rectangles or trapezoids for each method.

The script addresses the complexities that arise when approximating areas under a curve that crosses the x-axis, resulting in portions of the graph being above and below the x-axis. It explains that the height of the rectangles can be zero if the function value at an endpoint is at an x-intercept, leading to rectangles with no area. The script also discusses how to handle subintervals where the function is below the x-axis, using negative values for the heights of the rectangles and emphasizing the concept of signed area to account for regions below the x-axis.

The final part of the script makes a connection between the area approximation techniques and definite integrals, which are a fundamental concept in calculus. Using a table of values, the script demonstrates how to approximate the integral of f(x) from 0 to 25 using a midpoint sum with three subintervals of varying widths. It shows the process of calculating the function values at the midpoints, determining the signed areas for each subinterval, and summing these to approximate the integral. The script concludes by emphasizing that the area calculated can be negative, which is represented as the signed area, and that this signed area corresponds to the definite integral.

In the closing paragraph, the presenter thanks the viewers for joining and expresses her anticipation for the next lesson. She hints at further exploration of the notation and concepts introduced in the video, indicating that more detailed explanations will be provided in the second part of the lesson. The presenter's sign-off is warm and inviting, setting the stage for continued learning and engagement in future sessions.