How To Find The Area And Centroid (x, Y) Of The Region Bounded By The Curve Y² = X³, The X-axis, And The Line X=1?

Jun 21, 2025 by ADMIN 115 views

Unveiling Area and Centroid Calculation for y² = x³ Bounded Region

Introduction

In the realm of mathematics and engineering, determining geometric properties such as area and centroid is crucial for various applications. This article delves into the process of calculating the area and centroid (x, y) of a region bounded by the curve y² = x³, the x-axis, and the vertical line x = 1. We will explore the mathematical principles and steps involved in solving this problem, providing a comprehensive understanding of the concepts and techniques employed. Understanding the area and centroid of a shape is fundamental in various fields, including physics, engineering, and computer graphics. In physics, the centroid represents the center of mass of an object, while in engineering, it is crucial for structural analysis and design. The area calculation is equally important, as it can determine the amount of material needed, the load distribution, and other critical parameters. This article aims to provide a detailed explanation of the methods used to calculate these properties for a specific region bounded by a curve, an axis, and a line.

Problem Statement

Our primary goal is to determine the area and centroid (x̄, ȳ) of the region enclosed by the curve y² = x³, the x-axis (y = 0), and the vertical line x = 1. This region is defined within a coordinate system where the x-axis represents the horizontal direction and the y-axis represents the vertical direction. The curve y² = x³ is a parabolic curve, and the line x = 1 is a vertical line intersecting the x-axis at x = 1. The x-axis itself serves as the lower boundary of the region. The centroid, often referred to as the center of mass or center of gravity, is the geometric center of the region. It is the point where the region would perfectly balance if it were a thin plate of uniform density. The centroid is represented by its coordinates (x̄, ȳ), where x̄ is the x-coordinate of the centroid and ȳ is the y-coordinate of the centroid. Accurate determination of the centroid is crucial in applications such as structural mechanics, where it is used to analyze the stability and balance of structures. This article will walk through the steps necessary to calculate both the area and the centroid of the described region.

Visualizing the Region

Before diving into the calculations, it's essential to visualize the region we're working with. The curve y² = x³ creates a shape that starts at the origin (0,0) and extends to the right. Since y² = x³, we have two branches: y = x^(3/2) and y = -x^(3/2). These branches are symmetric about the x-axis. The region of interest is bounded above by the curve y = x^(3/2), below by the x-axis (y = 0), and to the right by the vertical line x = 1. This region looks like a curved triangle nestled between the x-axis and the curve, ending at the point (1,1). Understanding the visual representation of the region helps in setting up the integrals correctly and interpreting the results. For instance, the symmetry about the x-axis can simplify the calculation of the y-coordinate of the centroid. The line x = 1 acts as a clear boundary, defining the extent of the region along the x-axis. By visualizing the region, we can anticipate the general shape and location of the centroid, which serves as a sanity check for our calculations. The visualization also aids in identifying the limits of integration, which are crucial for computing the area and centroid coordinates. A clear mental image of the region is a valuable tool in solving this problem accurately.

Calculating the Area

To calculate the area (A) of the region, we'll use integral calculus. The area can be found by integrating the function y = x^(3/2) with respect to x from 0 to 1. This represents summing up the infinitesimally small vertical strips that make up the region. The area A is given by the integral: A = ∫[0 to 1] x^(3/2) dx. The limits of integration, 0 and 1, correspond to the x-values where the region starts and ends. The integrand, x^(3/2), represents the height of the region at each point x. The integral essentially calculates the cumulative height of the region over the interval [0, 1], giving us the total area. This method is a standard application of calculus in determining the area under a curve. It's important to correctly identify the integrand and the limits of integration to ensure an accurate calculation. The concept of dividing the area into infinitesimally small strips is a fundamental technique in calculus, allowing us to calculate areas of irregular shapes. The integration process then sums up these strips to give the total area, providing a powerful tool for solving geometric problems.

Performing the Integration

Now, let's evaluate the integral. The antiderivative of x^(3/2) is (2/5)x^(5/2). Applying the limits of integration, we have: A = [(2/5)(1)^(5/2)] - [(2/5)(0)^(5/2)] = (2/5) - 0 = 2/5. Therefore, the area of the region is 2/5 square units. This result represents the two-dimensional space enclosed by the curve, the x-axis, and the line x = 1. The calculation involves applying the power rule for integration, which is a fundamental technique in calculus. The antiderivative, (2/5)x^(5/2), is obtained by increasing the exponent of x by 1 and dividing by the new exponent. Evaluating the antiderivative at the upper and lower limits of integration gives us the definite integral, which represents the area. The units of the area are square units, reflecting the two-dimensional nature of the region. The result, 2/5 square units, is a numerical value that quantifies the size of the region. This area calculation is a crucial step in determining the centroid, as it is used in the formulas for the centroid coordinates.

Calculating the Centroid

To calculate the centroid (x̄, ȳ), we need to find the moments about the y-axis (Mx) and the x-axis (My). The coordinates of the centroid are then given by x̄ = My/A and ȳ = Mx/A, where A is the area we calculated earlier. The centroid represents the geometric center of the region, and its coordinates provide a precise location of this center. The moments Mx and My are measures of the distribution of the area about the respective axes. My represents the tendency of the area to rotate about the y-axis, while Mx represents the tendency to rotate about the x-axis. These moments are calculated using integrals that take into account the distance of each point in the region from the respective axis. The formulas x̄ = My/A and ȳ = Mx/A provide a way to determine the centroid coordinates from the moments and the area. These formulas are derived from the principles of mechanics and are widely used in engineering and physics to determine the center of mass of an object. The accurate calculation of the centroid is crucial in various applications, including structural analysis, where it is used to determine the stability and balance of structures.

Calculating My

The moment about the y-axis, My, is calculated by integrating x * y with respect to x from 0 to 1. Since y = x^(3/2), we have My = ∫[0 to 1] x * x^(3/2) dx = ∫[0 to 1] x^(5/2) dx. This integral represents the sum of the moments of infinitesimally small vertical strips about the y-axis. The integrand, x * x^(3/2) = x^(5/2), represents the moment of each strip, which is the product of its area (x^(3/2) dx) and its distance from the y-axis (x). The limits of integration, 0 and 1, correspond to the x-values where the region starts and ends. Evaluating this integral will give us the total moment of the region about the y-axis. This calculation is a crucial step in determining the x-coordinate of the centroid. The integral setup reflects the physical interpretation of the moment as the tendency of the area to rotate about the y-axis. The correct identification of the integrand and the limits of integration is essential for an accurate calculation of My.

Evaluating the Integral for My

The antiderivative of x^(5/2) is (2/7)x^(7/2). Applying the limits of integration, we get My = [(2/7)(1)^(7/2)] - [(2/7)(0)^(7/2)] = 2/7. Now, we can find x̄ using the formula x̄ = My/A. We already calculated A = 2/5, so x̄ = (2/7) / (2/5) = (2/7) * (5/2) = 5/7. Therefore, the x-coordinate of the centroid is 5/7. This result represents the horizontal position of the centroid relative to the y-axis. The calculation involves applying the power rule for integration, which is a fundamental technique in calculus. The antiderivative, (2/7)x^(7/2), is obtained by increasing the exponent of x by 1 and dividing by the new exponent. Evaluating the antiderivative at the upper and lower limits of integration gives us the definite integral, which represents the moment My. The division of My by the area A then yields the x-coordinate of the centroid. The value 5/7 indicates that the centroid is located slightly to the right of the midpoint of the region along the x-axis. This result is consistent with the shape of the region, which is skewed towards the right.

Calculating Mx

The moment about the x-axis, Mx, is calculated by integrating (1/2) * y² with respect to x from 0 to 1. Since y² = x³, we have Mx = ∫[0 to 1] (1/2) * x³ dx. This integral represents the sum of the moments of infinitesimally small horizontal strips about the x-axis. The integrand, (1/2) * y² = (1/2) * x³, represents the moment of each strip, which is half the square of its distance from the x-axis (y²) multiplied by its width (dx). The limits of integration, 0 and 1, correspond to the x-values where the region starts and ends. Evaluating this integral will give us the total moment of the region about the x-axis. This calculation is a crucial step in determining the y-coordinate of the centroid. The factor of 1/2 arises from the fact that we are considering the moment of a strip of infinitesimal width about the x-axis. The correct setup of this integral is essential for an accurate calculation of Mx.

Evaluating the Integral for Mx

The antiderivative of (1/2)x³ is (1/8)x⁴. Applying the limits of integration, we get Mx = [(1/8)(1)⁴] - [(1/8)(0)⁴] = 1/8. Now, we can find ȳ using the formula ȳ = Mx/A. We already calculated A = 2/5, so ȳ = (1/8) / (2/5) = (1/8) * (5/2) = 5/16. Therefore, the y-coordinate of the centroid is 5/16. This result represents the vertical position of the centroid relative to the x-axis. The calculation involves applying the power rule for integration, which is a fundamental technique in calculus. The antiderivative, (1/8)x⁴, is obtained by increasing the exponent of x by 1 and dividing by the new exponent and the constant factor. Evaluating the antiderivative at the upper and lower limits of integration gives us the definite integral, which represents the moment Mx. The division of Mx by the area A then yields the y-coordinate of the centroid. The value 5/16 indicates that the centroid is located below the midpoint of the region along the y-axis. This result is consistent with the shape of the region, which is concentrated closer to the x-axis.

Final Result

In conclusion, the area of the region bounded by the curve y² = x³, the x-axis, and the line x = 1 is 2/5 square units. The centroid (x̄, ȳ) of this region is located at (5/7, 5/16). This means that the geometric center of the region is at the point with coordinates x = 5/7 and y = 5/16. The area calculation provides a quantitative measure of the size of the region, while the centroid coordinates pinpoint its geometric center. These results are crucial in various applications, including structural analysis, where the centroid is used to determine the balance and stability of structures. The method used in this article involves the application of integral calculus, a powerful tool for calculating geometric properties of regions bounded by curves. The accurate determination of the area and centroid requires careful setup of the integrals and correct evaluation of the antiderivatives. The final results provide a complete characterization of the region, allowing for further analysis and application in various fields.

Practical Applications

The calculations performed in this article have significant practical applications in various fields. In engineering, the centroid is crucial for structural design, ensuring stability and balance. For example, when designing a bridge or a building, engineers need to know the centroid of various structural components to ensure that the load is distributed evenly. If the centroid is not properly accounted for, the structure may be unstable and prone to failure. The area calculation is also essential in engineering applications, such as determining the amount of material needed for a project. Knowing the area of a surface allows engineers to estimate the cost of materials and plan the construction process more efficiently. In physics, the centroid represents the center of mass of an object. This is the point where the object's mass is concentrated, and it is crucial for understanding the object's motion and stability. For example, when analyzing the trajectory of a projectile, physicists need to know the center of mass to accurately predict its path. The area calculation is also used in physics to determine the surface area of objects, which is important for calculating heat transfer, fluid flow, and other physical phenomena. In computer graphics, the centroid is used for object manipulation and rendering. Knowing the centroid of an object allows computer graphics programmers to easily rotate, scale, and position the object in a three-dimensional scene. The area calculation is also used in computer graphics to determine the size and shape of objects, which is important for creating realistic images. The methods discussed in this article provide a foundation for solving more complex problems in these and other fields.

Conclusion

In summary, we have successfully determined the area and centroid of the region bounded by the curve y² = x³, the x-axis, and the line x = 1. The area was found to be 2/5 square units, and the centroid was located at (5/7, 5/16). These results were obtained using integral calculus, a fundamental tool in mathematics and engineering. The process involved setting up and evaluating definite integrals to calculate the area and moments about the axes. The centroid coordinates were then determined using the formulas x̄ = My/A and ȳ = Mx/A. The accurate calculation of these geometric properties is essential in various applications, including structural analysis, mechanics, and computer graphics. The methods presented in this article provide a clear and concise approach to solving similar problems involving areas and centroids of regions bounded by curves. The understanding of these concepts and techniques is crucial for students and professionals in STEM fields. The ability to calculate areas and centroids allows engineers and physicists to analyze and design structures, predict the motion of objects, and create realistic computer graphics. The principles discussed in this article can be extended to more complex shapes and regions, making them a valuable tool in a wide range of applications. This article has provided a comprehensive explanation of the steps involved in calculating the area and centroid, equipping the reader with the knowledge and skills to tackle similar problems.