4.4 - Properties of Definite Integrals

This paragraph introduces the concept of definite integrals and discusses an additional property that allows for the integration of more complex functions. It explains how to handle functions with different rules over various parts of their domain by splitting the integral at an intermediate value within the limits of integration. This property is particularly useful for integrating piecewise functions, which have multiple rules that apply to different ranges of x-values. An example is provided to illustrate the process of finding the area under the curve of a function with two distinct rules by integrating separately over each rule's range and summing the results.

The paragraph delves into the process of evaluating definite integrals by finding anti-derivatives and applying them to specific intervals. It demonstrates how to integrate a function defined by two different rules over separate intervals and then combine these to find the total area under the curve. The example involves integrating from negative three to two and then from two to six, evaluating the anti-derivatives at the respective limits, and combining the results to find the total area, which is shown to be 83/3 square units.

This section explores the integration of absolute value functions and how the additive rule for integrals can be applied to these. The absolute value is defined piecewise, either as the number itself for non-negative inputs or as the negation of the number for negative inputs. An example is given for integrating the absolute value of (x - 3) from 0 to 4, which requires splitting the integral at x = 3 and simplifying the expression before integrating. The final area under the curve is calculated to be 5 square units.

The paragraph explains how to find the area enclosed between two curves using definite integrals. It introduces the concept of an upper and lower function and demonstrates how to calculate the area between them as the difference of their respective areas from a to b. The importance of identifying the correct upper and lower functions and their intersection points is emphasized. A general procedure for finding the area between two curves is outlined, which includes determining the points of intersection and the limits of integration.

This paragraph discusses methods for determining which function is the upper and which is the lower when finding the area between two curves. It suggests using a graphing calculator or a test value within the integration range to ascertain the functions' relative positions. An example is provided with two functions, f(x) and g(x), where the points of intersection are found algebraically and graphically, and the functions' roles as upper or lower are determined accordingly.

The paragraph focuses on algebraic techniques to find the points of intersection and to determine the upper and lower functions for calculating the area between curves. It shows how to set the functions equal to each other to find the x-values of intersection and how to choose a test value within the integration range to identify the upper function. The process is illustrated with an example where the functions are integrated from 0 to 2, and the area enclosed by the curves is calculated to be 4/3 square units.

This section addresses the scenario where two functions intersect at more than two points, leading to changes in the orientation of the graphs and the need to calculate the area in segments. The concept of negative area is introduced, and a strategy is presented for dealing with it by splitting the integral into parts that account for the changes in the upper and lower functions. An example demonstrates how to calculate the total area by summing the areas calculated over different intervals defined by the points of intersection.

The paragraph explains how to approach calculating the area when the orientation of the graphs changes within the interval of integration. It suggests breaking down the area into parts where the upper and lower functions can be consistently defined. This approach involves writing separate definite integrals for each part and then summing these to find the total area. The example provided involves integrating from negative five to three, with changes in the upper and lower functions at negative one, resulting in two separate integrals that are summed for the total area.

This part of the script discusses a method for addressing negative areas that may arise when integrating functions that cross the x-axis. It proposes treating the x-axis as a second function and adjusting the integral to ensure that all areas are considered as positive. The strategy involves breaking down the integral into segments based on the points where the function crosses the x-axis and then integrating each segment with the appropriate function as the upper or lower function. This approach ensures that the total area is calculated as a sum of positive values.

The paragraph introduces the concept of using definite integrals to find the average value of a function over an interval. It explains that the average value can be found by dividing the definite integral of the function by the width of the interval. An example is provided to illustrate the calculation of the average value of a function from 0 to 2. The concept is then applied to a real-world scenario where the average weekly sales of an online business are calculated for the first five weeks of operation using the definite integral of the sales function over the given interval.

This section demonstrates how to calculate the average weekly sales for a specific range of weeks, in this case, from week two to week five, using definite integrals. It clarifies the limits of integration, which start at one week (t=1) instead of zero, and end at five weeks (t=5). The calculation involves integrating the sales function from one to five and then adjusting the result to reflect the actual dollar amount in hundreds of dollars. The final step is to convert the result into an approximate dollar amount, providing a meaningful average weekly sales figure for the specified period.

The final paragraph recaps the topics covered in the discussion about area and integration, including Riemann sums, definite integrals, and strategies for finding areas between curves. It highlights the complexity of integration compared to differentiation and notes that while there are standard rules for many functions, there isn't always a definitive formula for more complicated functions. The paragraph previews upcoming topics that will explore strategies for integrating more complex functions, such as products of functions, nested functions, and compositions.