Calculus AB Lesson 7.3: Area Between Curves

This paragraph introduces the concept of calculating the area between two curves using definite integrals, reminiscent of Riemann sums. The focus is on understanding the width and height of rectangles that approximate the area under a curve, and the importance of identifying the right and left x-values for the width of these rectangles. The paragraph guides the learner to consider the vertical distance between the curves as the height of the rectangles and sets the stage for the development of the integral expression for the area.

The second paragraph delves deeper into the specifics of calculating the dimensions of rectangles used in approximating areas under curves. It explains how to find the width of each rectangle as ΔX, the change in X, and the height as the difference in Y values between two functions, f(X) and g(X). The area of each rectangle is then expressed as the product of its base (ΔX) and height [f(X_k) - g(X_k)]. The paragraph also encourages the learner to consider the sum of areas of all rectangles to approximate the total area between the curves.

Building upon the previous discussion, this paragraph introduces the summation of the areas of all rectangles to find the total area between two curves. It presents the transition from a Riemann sum to a definite integral, both conceptually and notationally, to represent the exact area. The paragraph explains the use of limits to approach infinity for the number of rectangles and introduces the integral notation with its lower and upper bounds to encapsulate the area between the curves.

The focus shifts to practical application as the paragraph discusses finding the area between two given functions by first determining their points of intersection, which define the interval for the definite integral. It illustrates the process of setting up integral expressions for the area between the curves by considering the top and bottom functions, f(X) and g(X), and emphasizes the importance of finding where the curves intersect to establish the correct limits for integration.

This paragraph explores the concept of top and bottom functions in the context of calculating areas between curves. It explains that the area is found by integrating the difference of the top function minus the bottom function over the interval defined by the points of intersection. The paragraph also discusses the importance of recognizing which function is above the other within the interval and how this affects the setup of the integral.

The paragraph encourages reflection on the methodology used for calculating areas between curves. It highlights the flexibility of the method, which can accommodate functions that are below the x-axis or intersect in various ways. The discussion emphasizes understanding the significance of vertical distances (top minus bottom) and horizontal distances (right minus left) in setting up integrals for areas, whether using vertical or horizontal rectangles.

This paragraph provides a detailed walkthrough of calculating areas between curves when the curves are not functions, requiring piecewise definitions. It discusses the challenges of using vertical rectangles in such cases and suggests using horizontal rectangles instead. The explanation includes how to adjust the notation and setup for horizontal rectangles, including identifying the width (ΔY) and height (right X minus left X) of the rectangles and setting up the integral with the correct limits and differential (dy).

The paragraph presents examples of setting up integrals for areas between curves using horizontal rectangles. It demonstrates how to identify the right and left X values as functions of Y, how to determine the correct interval for integration based on Y values, and how to express the area as the integral of the difference between the right and left X functions with respect to Y (dy). The examples illustrate the process of solving for X and setting up the integral correctly.

This paragraph extends the concept to calculate areas in regions bounded by multiple curves and axes. It describes the process of dividing the region into subregions, each requiring separate integrals due to different functions being on top or bottom within each subregion. The explanation includes identifying the correct functions for each subregion, setting up the integrals with appropriate limits, and adding the results to find the total area.

The final paragraph discusses the critical thinking required to decide whether to use vertical or horizontal slices for calculating areas between curves. It highlights situations where one method may be more straightforward than the other, such as when the same function touches both the top and bottom of a rectangle in the chosen orientation. The paragraph encourages the learner to consider the ease of setting up the integral and the clarity of the function relationships when making this choice.