Problem 51 - Calculus: One and Several Variables

Question

Find the centroid of the region under the curve \(y=\) \(\left(x^{2}+1\right)^{-1}\) from \(x=0\) to \(x=1\).

Step-by-Step Solution

Answer

The centroid of the region under the curve \(y = \frac{1}{x^2+1}\) from \(x=0\) to \(x=1\) is \(\left(\frac{2\ln(2)}{\pi}, \frac{1}{2}\right)\).

Step 1: Compute the Area under the Curve

First, we need to find the area under the curve \(y = \frac{1}{x^2+1}\) over the given range [0, 1]. This can be done using the definite integral: \[A = \int_0^1 \frac{1}{x^2+1}\, dx\]

Step 2: Evaluate the Integral for Area

For this step, we need to evaluate the integral: \[A = \int_0^1 \frac{1}{x^2+1}\, dx = [\arctan(x)]_0^1\] \[A = \arctan(1) - \arctan(0) = \frac{\pi}{4}\]

Step 3: Compute the x-coordinate of the Centroid

To find C_x, we need to compute the definite integral: \[C_x = \frac{1}{A} \int_0^1 x \cdot \frac{1}{x^2+1}\, dx\]

Step 4: Evaluate the Integral for x-coordinate

In this step, we will evaluate the integral: \[C_x = \frac{1}{\frac{\pi}{4}}\left[\frac{1}{2}\ln(x^2 + 1)\right]_0^1\] \[C_x = \frac{4}{\pi}\left[\frac{1}{2}\ln(2) - 0\right] = \frac{2\ln(2)}{\pi}\]

Step 5: Compute the y-coordinate of the Centroid

To find C_y, we need to compute the definite integral: \[C_y = \frac{1}{A} \int_0^1 \frac{y}{2}\, dx = \frac{1}{A} \int_0^1 \frac{1}{2(x^2+1)}\, dx\]

Step 6: Evaluate the Integral for y-coordinate

In this step, we will evaluate the integral: \[C_y = \frac{1}{\frac{\pi}{4}}\left[\frac{1}{4}\arctan(x)\right]_0^1\] \[C_y = \frac{4}{\pi}\left[\frac{1}{4}\frac{\pi}{4} - 0\right] = \frac{1}{2}\]

Step 7: Write down the Centroid Coordinates

Now that we have found the x-coordinate (\(C_x = \frac{2\ln(2)}{\pi}\)) and y-coordinate (\(C_y = \frac{1}{2}\)) of the centroid, we can write the centroid coordinates as: \[\textrm{Centroid} = \left(\frac{2\ln(2)}{\pi}, \frac{1}{2}\right)\]

Key Concepts

Understanding Definite Integrals

The definite integral is essential in calculus when you want to find the area under a curve over a specified interval. Think of it like summing up tiny vertical slices beneath the curve between two x-values. This process helps determine the total accumulated change or area. Also, it forms the foundation for determining more complex attributes, like centroids.

For instance, to find the area under the curve of the function \(y = \frac{1}{x^2+1}\) from \(x = 0\) to \(x = 1\), you'll compute the definite integral. This is visually akin to laying a 2D shape over the curve and observing how much space it covers beneath it.

Computing this definite integral helps identify crucial physical properties related to the curve, such as the total area and the average height of the curve across the specified interval. Definite integrals are useful in many scientific fields, such as physics and engineering, where knowing the total quantity represented by a given range is essential.

Exploring Integration Techniques

Integration, a core technique in calculus, involves finding functions that describe the accumulation of quantities. There are various methods and approaches used to solve integrals, depending on the function form.

Common techniques include:

• Substitution Method: Use when integral contains a composite function. Change variables to simplify.
• Integration by Parts: Break down integral of a product into simpler parts.
• Partial Fraction Decomposition: Decompose complex fractions to simpler ones, useful for rational functions.

In solving \(\int_0^1 \frac{1}{x^2+1}\, dx\), recognizing the function as an inverse trigonometric function, specifically arctan, is key. Noticing these connections is crucial in choosing an efficient method.

Practicing various techniques sharpens your problem-solving skills and builds intuition for tackling diverse calculus problems. It's like having a toolbox where you pick just the right tool for a certain type of screw or nail, matching the right technique to the integral.

Approaching Calculus Problems

Solving calculus problems can feel daunting, but by steadily honing a few core strategies, you can simplify the process. It doesn't only involve getting the correct answer but understanding the path taken to arrive there.

When addressing problems requiring the computation of a centroid, for example, break down the problem into smaller steps:

• Compute the area under the curve using definite integrals.
• Find the x-coordinate of the centroid through integration by focusing on weighting the position.
• Calculate the y-coordinate by assessing the average height.
• Simplify your results ensuring you've factored in any constants correctly.

By breaking it down, solving calculus problems becomes manageable. Each part has its logic and requires careful attention. Remember, practice builds understanding and expertise, so take each problem as a chance to learn something new or reinforce what you already know. Each time you solve a problem, you add another layer of knowledge that helps tackle even more challenging questions in the future.