Area between three curves is a fundamental concept in calculus that deals with finding the area enclosed by three different functions on a given interval. This concept is widely used in various fields such as physics, engineering, and economics to analyze and solve real-world problems. In this article, we will explore the methods and techniques used to calculate the area between three curves, along with some practical examples.
The area between three curves can be visualized as the region enclosed by the curves on a graph. To calculate this area, we need to determine the points of intersection between the curves and then integrate the functions over the interval that defines the area. There are two main methods to calculate the area between three curves: the vertical method and the horizontal method.
The vertical method involves integrating the functions with respect to the vertical axis. First, we find the points of intersection between the curves by setting them equal to each other. Then, we integrate the functions from the lower to the upper limits of the interval, taking into account the signs of the functions to determine the area between them. If the function is above the x-axis, the area is positive; if it is below the x-axis, the area is negative.
For example, consider the functions f(x) = x^2, g(x) = x, and h(x) = 0. To find the area between these curves on the interval [0, 2], we first find the points of intersection by setting f(x) = g(x) and g(x) = h(x). This gives us x = 0 and x = 1 as the points of intersection. Next, we integrate the functions over the interval [0, 2] as follows:
Area = ∫[0, 2] (g(x) – f(x)) dx = ∫[0, 2] (x – x^2) dx = [x^2/2 – x^3/3] from 0 to 2 = 2/2 – 8/3 = 1/3
Therefore, the area between the three curves on the interval [0, 2] is 1/3 square units.
The horizontal method involves integrating the functions with respect to the horizontal axis. This method is useful when the functions are not easily integrated with respect to the vertical axis. To use the horizontal method, we need to express the functions in terms of y and then find the points of intersection. After that, we integrate the functions over the interval that defines the area, taking into account the signs of the functions.
For instance, let’s consider the functions f(y) = y^2, g(y) = y, and h(y) = 0. To find the area between these curves on the interval [0, 2], we first express the functions in terms of y and find the points of intersection. This gives us y = 0 and y = 1 as the points of intersection. Next, we integrate the functions over the interval [0, 2] as follows:
Area = ∫[0, 2] (g(y) – f(y)) dy = ∫[0, 2] (y – y^2) dy = [y^2/2 – y^3/3] from 0 to 2 = 2/2 – 8/3 = 1/3
Thus, the area between the three curves on the interval [0, 2] is also 1/3 square units, as we found using the vertical method.
In conclusion, the area between three curves is an essential concept in calculus that can be calculated using either the vertical or horizontal method. By understanding the methods and techniques involved, we can solve a wide range of real-world problems in various fields.
