Calculate Arc Length Of Y=x^2/4 - Ln(x/2) | Algebra Problem

Hey everyone! Today, we're diving into a fun problem where we'll calculate the arc length of a curve. Specifically, we're dealing with the curve defined by the equation y = x^2/4 - ln(x/2), and we want to find its length over the interval 1 ≤ x ≤ e. Buckle up, because we're about to put our calculus hats on and get to work!

Understanding Arc Length

Before we jump into the nitty-gritty, let's take a moment to understand what we mean by "arc length." Imagine you have a curvy road on a map. The arc length is simply the distance you'd travel if you drove along that road. In mathematical terms, it's the length of the curve between two points. To find this length, we'll use a formula derived from calculus, which essentially breaks the curve into tiny, almost straight segments and adds up their lengths. This might sound a bit intimidating, but don't worry, we'll break it down step by step.

The arc length formula is the heart of our solution. It tells us exactly how to calculate the length of a curve given its equation and the interval over which we want to measure. This formula is derived using the Pythagorean theorem and a bit of calculus magic, but the result is quite elegant and powerful. The formula states that the length L of a curve y(x) from x = a to x = b is given by the integral: L = ∫[a, b] √(1 + (y'(x))^2) dx. Here, y'(x) represents the derivative of y(x) with respect to x, and the integral sums up the infinitesimal arc lengths along the curve. This formula is a cornerstone of calculus and is used extensively in various fields, including physics, engineering, and computer graphics. Understanding this formula is crucial for tackling problems involving curves and their lengths.

So, why is this formula so important? Well, it allows us to measure the length of curves that aren't simple straight lines. Think about the path of a roller coaster, the shape of a suspension bridge cable, or even the trajectory of a projectile. These are all curves, and the arc length formula provides a way to quantify their lengths. Moreover, the concept of arc length extends beyond simple curves in a plane. It can be generalized to curves in three-dimensional space and even to higher dimensions. This makes it a versatile tool for analyzing complex shapes and paths. For example, in computer graphics, arc length is used to parameterize curves, which is essential for rendering smooth and realistic shapes. In physics, it helps in calculating the distance traveled by an object along a curved path. In engineering, it's used to design structures that can withstand the forces acting on them. The applications are vast and varied, making the arc length formula a fundamental concept to grasp.

Step-by-Step Solution

Let's dive into the solution step-by-step. This will help us understand how to apply the arc length formula and get to the final answer. Don't worry if some steps seem tricky at first, we'll break them down and explain the reasoning behind each one. By following along, you'll not only solve this particular problem but also gain a deeper understanding of how to tackle similar problems in the future.

1. Find the Derivative y'(x)

The first thing we need to do is find the derivative of our function, y(x) = x^2/4 - ln(x/2). Remember, the derivative represents the slope of the tangent line to the curve at any point. This is crucial for the arc length formula because it helps us determine the infinitesimal changes in the y-direction as we move along the curve. To find the derivative, we'll use the power rule and the chain rule of differentiation.

The power rule states that the derivative of x^n is nx^(n-1). So, the derivative of x^2/4 is (2x)/4 = x/2. For the second term, ln(x/2), we'll use the chain rule. The chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x). In our case, f(u) = ln(u) and g(x) = x/2. The derivative of ln(u) is 1/u, and the derivative of x/2 is 1/2. Applying the chain rule, the derivative of ln(x/2) is (1/(x/2)) * (1/2) = 1/x. Now, combining the derivatives of both terms, we get y'(x) = x/2 - 1/x. This is the slope of our curve at any point x, which is a key ingredient in our arc length calculation.

2. Square the Derivative (y'(x))^2

Next up, we need to square the derivative we just found. Squaring the derivative is a critical step because it appears inside the square root in the arc length formula. This step effectively converts the slope into a measure of the rate of change of the curve's length, regardless of whether the curve is increasing or decreasing. When we square y'(x), we're essentially finding a term that represents the square of the infinitesimal length element along the curve.

So, let's square y'(x) = x/2 - 1/x. We have (y'(x))^2 = (x/2 - 1/x)^2. Expanding this expression, we get (x/2)^2 - 2(x/2)(1/x) + (1/x)^2. Simplifying each term, we have x^2/4 - 1 + 1/x^2. This expression represents the square of the slope of our curve and is a crucial component in the arc length calculation. Squaring the derivative ensures that we are dealing with positive quantities, which is essential when we take the square root in the next step.

3. Add 1 and Take the Square Root √(1 + (y'(x))^2)

Now, we're getting closer to the integral! We need to add 1 to the squared derivative and then take the square root. This step is directly derived from the Pythagorean theorem, which forms the basis of the arc length formula. Adding 1 and taking the square root essentially calculates the length of the hypotenuse of a tiny right triangle formed by the infinitesimal changes in x and y along the curve. This hypotenuse represents the infinitesimal arc length, and we'll integrate it over the interval to find the total arc length.

So, let's add 1 to our previous result: 1 + (y'(x))^2 = 1 + (x^2/4 - 1 + 1/x^2). Simplifying, we get x^2/4 + 1/x^2. Notice something interesting? This expression looks like a perfect square! In fact, it's (x/2 + 1/x)^2. Now, we take the square root: √(1 + (y'(x))^2) = √((x/2 + 1/x)^2) = |x/2 + 1/x|. Since our interval is 1 ≤ x ≤ e, x is always positive, so we can drop the absolute value signs. Thus, we have √(1 + (y'(x))^2) = x/2 + 1/x. This simplified expression is what we'll integrate to find the arc length.

4. Evaluate the Integral ∫[1, e] √(1 + (y'(x))^2) dx

Alright, the moment we've been waiting for! Now we integrate the expression we just found over the interval [1, e]. This integral represents the sum of all the infinitesimal arc lengths along the curve from x = 1 to x = e. Evaluating this integral will give us the total arc length of the curve over the specified interval.

So, let's set up the integral: ∫[1, e] (x/2 + 1/x) dx. To evaluate this integral, we'll find the antiderivative of x/2 + 1/x. The antiderivative of x/2 is (x^2)/4, and the antiderivative of 1/x is ln|x|. So, the antiderivative of x/2 + 1/x is (x^2)/4 + ln|x|. Now, we evaluate this antiderivative at the limits of integration, e and 1, and subtract the results:

[(e^2)/4 + ln(e)] - [(1^2)/4 + ln(1)] = (e^2)/4 + 1 - 1/4 - 0 = (e^2)/4 + 3/4. This is the exact value of the arc length. Now, we need to approximate this value to one decimal place.

5. Approximate the Result

Finally, we need to approximate our result to one decimal place. We found that the arc length is (e^2)/4 + 3/4. Now, we'll plug in the value of e (approximately 2.71828) and calculate the result.

(e^2)/4 + 3/4 ≈ (2.71828^2)/4 + 3/4 ≈ 7.38905/4 + 0.75 ≈ 1.84726 + 0.75 ≈ 2.59726. Rounding this to one decimal place, we get 2.6. So, the arc length of the curve y = x^2/4 - ln(x/2) from x = 1 to x = e is approximately 2.6 units.

Conclusion

And there you have it! We've successfully calculated the arc length of the curve y = x^2/4 - ln(x/2) from x = 1 to x = e. The process involved finding the derivative, squaring it, adding 1, taking the square root, and finally, evaluating the integral. We then approximated the result to one decimal place, arriving at the answer of approximately 2.6. This problem showcases the power of calculus in solving real-world geometric problems. The arc length formula is a versatile tool that can be applied to a wide range of curves, making it a valuable concept to understand.

So, the next time you encounter a curvy path, remember the arc length formula, and you'll be able to measure its length with confidence. Keep practicing, and you'll become a master of calculus in no time!

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