Understanding the Area Between Two Curves Formula
The area between two curves is a fundamental concept in calculus that helps us calculate the space enclosed by two functions. This concept is widely used in various fields, including physics, engineering, and economics. The area between two curves formula is a powerful tool that allows us to determine the region enclosed by two functions over a specified interval.
The area between two curves formula is given by:
\[ A = \int_{a}^{b} |f(x) – g(x)| \, dx \]
where \( A \) represents the area between the curves, \( f(x) \) and \( g(x) \) are the two functions, and \( a \) and \( b \) are the lower and upper limits of integration, respectively. The absolute value in the formula ensures that the area is always positive, regardless of the order of the functions.
To understand the area between two curves formula, let’s consider an example. Suppose we have two functions, \( f(x) = x^2 \) and \( g(x) = x \), and we want to find the area between them over the interval \([0, 2]\).
First, we need to determine the points of intersection between the two functions. By setting \( f(x) = g(x) \), we get:
\[ x^2 = x \]
\[ x^2 – x = 0 \]
\[ x(x – 1) = 0 \]
This gives us two points of intersection: \( x = 0 \) and \( x = 1 \). Now, we can split the interval \([0, 2]\) into two subintervals: \([0, 1]\) and \([1, 2]\).
For the subinterval \([0, 1]\), \( f(x) \) is greater than \( g(x) \), so we can rewrite the area formula as:
\[ A_1 = \int_{0}^{1} (f(x) – g(x)) \, dx \]
\[ A_1 = \int_{0}^{1} (x^2 – x) \, dx \]
For the subinterval \([1, 2]\), \( g(x) \) is greater than \( f(x) \), so we can rewrite the area formula as:
\[ A_2 = \int_{1}^{2} (g(x) – f(x)) \, dx \]
\[ A_2 = \int_{1}^{2} (x – x^2) \, dx \]
Now, we can evaluate both integrals to find the area between the two curves:
\[ A_1 = \left[ \frac{x^3}{3} – \frac{x^2}{2} \right]_{0}^{1} = \frac{1}{3} – \frac{1}{2} = -\frac{1}{6} \]
\[ A_2 = \left[ \frac{x^2}{2} – \frac{x^3}{3} \right]_{1}^{2} = \frac{2}{2} – \frac{8}{3} – \left( \frac{1}{2} – \frac{1}{3} \right) = \frac{1}{6} \]
Finally, we can sum up the areas of both subintervals to find the total area between the two curves:
\[ A = A_1 + A_2 = -\frac{1}{6} + \frac{1}{6} = 0 \]
In this example, the area between the two curves is zero, which means that the curves overlap over the interval \([0, 2]\).
In conclusion, the area between two curves formula is a valuable tool for calculating the space enclosed by two functions. By understanding the formula and applying it to various examples, we can gain a deeper insight into the concept of area between two curves and its applications in different fields.
