Calculate Area Between Curves F(x) = 5 - Ln(x) And G(x) = X/e + 3
Finding the area of a region bounded by curves is a fundamental concept in calculus with wide-ranging applications in physics, engineering, economics, and other fields. This article delves into the process of calculating the area A of the region enclosed between two curves, f(x) = 5 - ln(x) and g(x) = x/e + 3, over the interval [1, 4]. We'll explore the underlying principles, the steps involved in setting up the integral, and the techniques for evaluating it to arrive at the exact answer. Let's embark on this mathematical journey to unravel the intricacies of area calculation between curves.
Understanding the Concept of Area Between Curves
Before diving into the specific problem, it's crucial to grasp the fundamental concept of finding the area between curves. Area between curves represents the amount of space enclosed by two or more curves within a defined interval. Imagine two curves, f(x) and g(x), where f(x) ≥ g(x) over the interval [a, b]. The area between these curves is essentially the integral of the difference between the two functions over that interval.
This concept stems from the idea of approximating the area using infinitesimally thin rectangles. Each rectangle has a width of dx (an infinitesimally small change in x) and a height equal to the difference between the function values, f(x) - g(x). Summing the areas of all these rectangles, as the width approaches zero, gives us the exact area between the curves. This summation process is precisely what the definite integral represents.
Mathematically, the area A between the curves f(x) and g(x) over the interval [a, b] is given by:
A = ∫[a, b] (f(x) - g(x)) dx
This formula is the cornerstone of our problem-solving approach. It encapsulates the essence of calculating the area between curves by integrating the difference in their function values. The key is to correctly identify which function is above the other within the given interval, as this determines the order of subtraction in the integrand. If the curves intersect within the interval, we may need to split the integral into multiple parts, ensuring that we always subtract the lower function from the upper function in each sub-interval.
In our case, we have f(x) = 5 - ln(x) and g(x) = x/e + 3, and we're interested in the interval [1, 4]. Before setting up the integral, it's essential to determine which function is greater than the other within this interval. This can be done by analyzing the graphs of the functions or by evaluating them at specific points within the interval. Once we know the relative positions of the curves, we can confidently set up the definite integral and proceed with its evaluation.
Setting Up the Integral for the Given Problem
Now, let's apply the concept of area between curves to our specific problem. We have the functions f(x) = 5 - ln(x) and g(x) = x/e + 3, and we want to find the area between these curves over the interval [1, 4]. The first crucial step is to determine which function is greater than the other within this interval.
We can analyze the behavior of the functions by considering their values at the endpoints of the interval and at any critical points within the interval. Let's evaluate f(x) and g(x) at x = 1 and x = 4:
- At x = 1:
- f(1) = 5 - ln(1) = 5 - 0 = 5
- g(1) = 1/e + 3 ≈ 0.368 + 3 = 3.368
- At x = 4:
- f(4) = 5 - ln(4) ≈ 5 - 1.386 = 3.614
- g(4) = 4/e + 3 ≈ 1.472 + 3 = 4.472
From these evaluations, we observe that f(x) > g(x) at x = 1, but g(x) > f(x) at x = 4. This suggests that the curves may intersect within the interval [1, 4]. To confirm this, we need to find the point(s) where f(x) = g(x). This involves solving the equation:
5 - ln(x) = x/e + 3
2 - ln(x) = x/e
This equation is transcendental and cannot be solved algebraically. We would typically resort to numerical methods (such as the Newton-Raphson method) or graphical analysis to find the intersection point. For the purpose of this explanation, let's assume that the curves intersect at a point x = c within the interval [1, 4]. This means we need to split the integral into two parts:
- From 1 to c, where f(x) ≥ g(x)
- From c to 4, where g(x) ≥ f(x)
Therefore, the area A can be expressed as the sum of two definite integrals:
A = ∫[1, c] (f(x) - g(x)) dx + ∫[c, 4] (g(x) - f(x)) dx
A = ∫[1, c] (5 - ln(x) - (x/e + 3)) dx + ∫[c, 4] (x/e + 3 - (5 - ln(x))) dx
A = ∫[1, c] (2 - ln(x) - x/e) dx + ∫[c, 4] (x/e - 2 + ln(x)) dx
This is the integral setup we need to evaluate to find the area between the curves. The next step is to find the intersection point c (using numerical methods if necessary) and then evaluate these definite integrals.
Evaluating the Definite Integrals
Now that we have set up the integrals, the next step is to evaluate them. This involves finding the antiderivatives of the integrands and applying the Fundamental Theorem of Calculus. Let's consider the first integral:
∫[1, c] (2 - ln(x) - x/e) dx
We need to find the antiderivative of each term separately:
- The antiderivative of 2 is 2x.
- The antiderivative of -ln(x) is -xln(x) + x (using integration by parts).
- The antiderivative of -x/e is -x^2/(2e).
Therefore, the antiderivative of the entire integrand is:
F(x) = 2x - xln(x) + x - x^2/(2e) = 3x - xln(x) - x^2/(2e)
Applying the Fundamental Theorem of Calculus, we evaluate F(x) at the limits of integration, 1 and c:
∫[1, c] (2 - ln(x) - x/e) dx = F(c) - F(1)
= (3c - cln(c) - c^2/(2e)) - (3(1) - 1ln(1) - 1^2/(2e))
= 3c - cln(c) - c^2/(2e) - 3 + 0 + 1/(2e)
Similarly, let's consider the second integral:
∫[c, 4] (x/e - 2 + ln(x)) dx
The antiderivative of each term is:
- The antiderivative of x/e is x^2/(2e).
- The antiderivative of -2 is -2x.
- The antiderivative of ln(x) is xln(x) - x (using integration by parts).
Therefore, the antiderivative of the entire integrand is:
G(x) = x^2/(2e) - 2x + xln(x) - x = x^2/(2e) - 3x + xln(x)
Applying the Fundamental Theorem of Calculus, we evaluate G(x) at the limits of integration, c and 4:
∫[c, 4] (x/e - 2 + ln(x)) dx = G(4) - G(c)
= (4^2/(2e) - 3(4) + 4ln(4)) - (c^2/(2e) - 3c + cln(c))
= 8/e - 12 + 4ln(4) - c^2/(2e) + 3c - cln(c)
Now, we add the results of the two integrals to find the total area A:
A = (3c - cln(c) - c^2/(2e) - 3 + 1/(2e)) + (8/e - 12 + 4ln(4) - c^2/(2e) + 3c - cln(c))
A = 6c - 2cln(c) - c^2/e - 15 + 1/(2e) + 8/e + 4ln(4)
A = 6c - 2cln(c) - c^2/e - 15 + 17/(2e) + 4ln(4)
To obtain the exact answer, we need to substitute the value of c (the intersection point) into this expression. As mentioned earlier, finding c typically requires numerical methods. Once we have the value of c, we can plug it into the equation above to calculate the area A.
Numerical Approximation and Final Answer
Since we cannot find an exact algebraic solution for the intersection point c, we would need to use a numerical method (e.g., Newton-Raphson method) to approximate its value. Let's assume, for the sake of demonstration, that the numerical method gives us an approximate value of c ≈ 2.2. Plugging this value into the area expression we derived earlier:
A ≈ 6(2.2) - 2(2.2)ln(2.2) - (2.2)^2/e - 15 + 17/(2e) + 4ln(4)
A ≈ 13.2 - 2(2.2)(0.788) - 4.84/2.718 - 15 + 17/(2*2.718) + 4(1.386)
A ≈ 13.2 - 3.47 - 1.78 - 15 + 3.13 + 5.54
A ≈ 2.62
Therefore, the approximate area between the curves over the interval [1, 4] is 2.62 square units. However, it's crucial to remember that this is an approximation based on the assumed value of c. To obtain a more accurate answer, a more precise numerical method should be employed to find the intersection point c.
In conclusion, finding the area between curves involves setting up a definite integral that represents the difference between the functions over the given interval. This often requires determining the intersection points of the curves and splitting the integral into multiple parts if the curves intersect within the interval. Evaluating the definite integrals using the Fundamental Theorem of Calculus yields the area between the curves. In cases where an exact algebraic solution is not possible, numerical methods can be used to approximate the intersection points and the area.