Finding Area Between Curves A Calculus Guide

In calculus, one of the most fundamental applications of integration is finding the area of a region bounded by curves. This concept has wide-ranging applications in various fields, including physics, engineering, economics, and computer graphics. In this comprehensive guide, we will delve into the methods and techniques for calculating the area between two curves using integration, providing a step-by-step approach with examples and explanations to solidify your understanding. Let's consider the specific scenario where we need to find the area of the region R bounded by two functions, f(x) and g(x), over a given interval [a, b]. To effectively calculate this area, we must first understand the underlying principles and techniques involved. The core concept behind finding the area between two curves lies in the idea of approximating the region with infinitesimally thin rectangles and then summing up their areas using integration. This approach allows us to handle complex shapes and curves that would be difficult to analyze using traditional geometric methods. By breaking down the region into small rectangular strips, we can approximate the area of each strip and then use integration to find the exact area of the entire region. This method is particularly useful when dealing with functions that are not easily represented by simple geometric shapes. Before we dive into the specific example provided, let's explore the general formula and methodology for finding the area between two curves. This will provide a solid foundation for understanding the problem at hand and tackling similar problems in the future.

Understanding the General Formula

The area A of the region R bounded by the curves y = f(x) and y = g(x), and the vertical lines x = a and x = b, where f(x) ≥ g(x) for all x in [a, b], is given by the definite integral:

A = ∫[a, b] (f(x) - g(x)) dx

This formula is based on the fundamental principle of calculus, which states that the definite integral of a function represents the area under its curve. When we subtract g(x) from f(x), we are essentially finding the vertical distance between the two curves at each point x. This vertical distance represents the height of an infinitesimally thin rectangle, and the integral sums up the areas of all such rectangles between x = a and x = b. It's crucial to note that f(x) must be greater than or equal to g(x) over the interval [a, b] for this formula to hold true. If the curves intersect within the interval, we may need to divide the interval into subintervals and apply the formula separately for each subinterval, ensuring that we always subtract the lower function from the upper function. The absolute value can also be used to simplify this process, as we will see later in the discussion. Understanding this general formula is the first step towards solving the specific problem at hand. By grasping the underlying principles, we can confidently apply the formula to various scenarios and accurately calculate the area between curves. Now, let's move on to the specific example and see how this formula can be applied in practice.

Applying the Formula to the Given Problem

In our specific problem, we are given the functions f(x) = (5x/4) + 2 and g(x) = (2x/3) - 8, and the interval [a, b] is [0, 5]. Our task is to represent the area A of the region R bounded by these functions over this interval by writing an integral with respect to x. To do this, we need to follow these steps:

1. Verify that f(x) ≥ g(x) over the interval [0, 5]: This is a crucial step because the formula A = ∫[a, b] (f(x) - g(x)) dx assumes that f(x) is always greater than or equal to g(x) within the interval. To verify this, we can either graph the functions or analyze their behavior algebraically. Let's first try to understand this graphically. The function f(x) = (5x/4) + 2 represents a straight line with a positive slope, and the function g(x) = (2x/3) - 8 also represents a straight line but with a smaller positive slope and a negative y-intercept. Intuitively, we can see that f(x) starts higher than g(x) at x = 0 and increases faster than g(x) as x increases. To confirm this algebraically, we can compare the values of f(x) and g(x) at the endpoints of the interval and at any critical points where the two functions might intersect. At x = 0, f(0) = 2 and g(0) = -8, so f(0) > g(0). At x = 5, f(5) = (55/4) + 2 = 8.25* and g(5) = (25/3) - 8 = -4.67*, so f(5) > g(5). These calculations suggest that f(x) ≥ g(x) over the interval [0, 5]. To be absolutely sure, we can find the points of intersection by setting f(x) = g(x) and solving for x. If there are no solutions within the interval [0, 5], it confirms that f(x) is always greater than g(x) within this interval.
2. Set up the integral: Once we have confirmed that f(x) ≥ g(x), we can set up the integral using the formula. The integral will represent the area A as the definite integral from a to b of the difference between f(x) and g(x). In our case, a = 0 and b = 5, so we will integrate from 0 to 5. The integrand will be f(x) - g(x), which is the difference between the two functions. Substituting the given functions, we get f(x) - g(x) = ((5x/4) + 2) - ((2x/3) - 8). This expression simplifies to (7x/12) + 10. Therefore, the integral that represents the area A is given by ∫[0, 5] ((5x/4) + 2 - (2x/3) + 8) dx. This integral represents the exact area of the region bounded by the curves f(x) and g(x) over the interval [0, 5]. It captures the essence of the area calculation by summing up the areas of infinitesimally thin rectangles between the two curves.
3. Simplify the integrand: Before writing the final answer, it's good practice to simplify the integrand as much as possible. This makes the integral easier to evaluate in the next step, if we were asked to compute the actual area. In our case, the integrand is ((5x/4) + 2 - (2x/3) + 8). Combining the like terms, we get ((5x/4) - (2x/3) + 2 + 8), which simplifies to ((15x - 8x)/12 + 10), and further simplifies to (7x/12 + 10). This simplified integrand will make the integration process more straightforward when we evaluate the integral.
4. Write the final answer: Based on the steps above, we can now write the integral that represents the area A. The integral is ∫[0, 5] ((7x/12) + 10) dx. This integral provides a concise and accurate representation of the area A. It encapsulates all the necessary information, including the functions, the interval, and the operation of integration. This is the final answer to the problem, representing the area A of the region R bounded by the given functions over the specified interval. The next step, if required, would be to evaluate this integral to find the numerical value of the area.

Representing the Area A

Based on the steps outlined above, the area A of the region R can be represented by the following definite integral:

A = ∫[0, 5] ((5x/4) + 2 - (2x/3) + 8) dx

Simplifying the integrand, we get:

A = ∫[0, 5] (7x/12 + 10) dx

This integral represents the exact area of the region R. It is a concise and precise mathematical expression that captures the essence of the area calculation. This representation allows us to further evaluate the integral to find the numerical value of the area, if required. In the next section, we will discuss the evaluation of this integral and the interpretation of the result.

Evaluating the Integral (Optional)

While the problem only asked for the integral representation, let's take it a step further and evaluate the integral to find the actual area. This will provide a more complete understanding of the problem and its solution. To evaluate the integral ∫[0, 5] (7x/12 + 10) dx, we can use the power rule for integration and the linearity of the integral. The power rule states that the integral of x^n is (x^(n+1))/(n+1), and the linearity of the integral allows us to integrate term by term. Applying these rules, we get:

∫[0, 5] (7x/12 + 10) dx = (7/12)∫[0, 5] x dx + 10∫[0, 5] dx

Now, we can integrate x and 1 separately. The integral of x is (x^2)/2, and the integral of 1 is x. So, we have:

(7/12)∫[0, 5] x dx + 10∫[0, 5] dx = (7/12) [(x^2)/2]_0^5 + 10[x]_0^5

Next, we evaluate the expressions at the limits of integration, 5 and 0. This means we substitute x = 5 and x = 0 into the expressions and subtract the results. Doing this, we get:

(7/12) [(5^2)/2 - (0^2)/2] + 10[5 - 0] = (7/12) (25/2) + 10(5)

Now, we simplify the expression:

(7/12) (25/2) + 10(5) = 175/24 + 50

To add these terms, we need a common denominator, which is 24. So, we convert 50 to a fraction with a denominator of 24:

50 = (50 * 24)/24 = 1200/24

Now we can add the fractions:

175/24 + 1200/24 = 1375/24

Therefore, the value of the integral is 1375/24, which is approximately 57.29. This means that the area of the region R bounded by the curves f(x) and g(x) over the interval [0, 5] is approximately 57.29 square units. This numerical value provides a concrete understanding of the size of the region. It is the exact area enclosed between the two curves within the specified interval. This step of evaluating the integral completes the problem and gives us a tangible result.

Conclusion

In this comprehensive guide, we have explored the method of finding the area between two curves using integration. We started with the general formula for calculating the area between two curves and then applied it to a specific problem. We learned how to verify that one function is greater than or equal to the other over the given interval, set up the integral, simplify the integrand, and represent the area as a definite integral. Additionally, we went a step further and evaluated the integral to find the numerical value of the area. This detailed approach provides a solid understanding of the concepts and techniques involved in finding the area between curves using integration. By mastering these methods, you will be well-equipped to tackle a wide range of calculus problems and apply these concepts to real-world scenarios. The ability to find the area between curves is a valuable tool in various fields, and a thorough understanding of this topic is essential for success in calculus and beyond. Remember, practice is key to mastering these concepts, so be sure to work through additional examples and problems to solidify your understanding. The more you practice, the more confident you will become in your ability to apply these techniques and solve complex problems.

Keywords: area between curves, integration, definite integral, functions f(x) and g(x), interval [a, b], calculus, area calculation, integral representation, evaluating integrals, mathematical expression.

Express the area A of the region R bounded by the functions ${f(x) = \frac{5x}{4} + 2}$ and ${g(x) = \frac{2x}{3} - 8}$ over the interval ${0, 5}$ as an integral with respect to x.

Finding Area Between Curves A Calculus Guide