Surface Area Revolution Find Solid Formed By Revolving Y=e^(-x^2)

by ADMIN 66 views

Delving into the fascinating realm of calculus, we embark on a journey to explore the surface area of a solid generated by revolving a curve around an axis. Our specific focus lies on the function y = e(-x2), a captivating Gaussian function, defined over the interval [-3, 3]. We shall meticulously dissect the process of revolving this curve around the x-axis and then masterfully calculate the surface area of the resulting solid, a task that demands precision and an intimate understanding of integral calculus.

Grasping the Essence of Surface Area of Revolution

Before we plunge into the intricacies of our specific problem, let's first establish a firm grasp of the fundamental concept of surface area of revolution. Imagine taking a curve defined by a function y = f(x) over an interval [a, b] and revolving it around the x-axis. This revolution generates a three-dimensional solid, and our primary objective is to determine the area of the curved surface of this solid. To achieve this, we employ the powerful tool of integral calculus, which allows us to break down the complex surface into infinitesimal elements, calculate their individual areas, and then sum them up through integration. This process hinges on the surface area formula of revolution, a cornerstone of calculus that empowers us to solve a wide range of problems involving curved surfaces.

Laying the Foundation: The Surface Area Formula

The surface area (SA) of a solid formed by revolving the curve y = f(x) over the interval [a, b] around the x-axis is given by the following integral:

SA = 2π ∫[a, b] y √(1 + (dy/dx)²) dx

This formula, seemingly complex at first glance, is built upon a simple yet elegant principle. It essentially integrates the product of the circumference of a circle (2πy) and an infinitesimal arc length (√(1 + (dy/dx)²) dx) along the curve. The term √(1 + (dy/dx)²) represents the arc length element, which approximates the length of a small segment of the curve. By integrating this product over the interval [a, b], we effectively sum up the areas of infinitesimally thin bands that make up the entire surface of the solid.

Deconstructing the Formula: A Step-by-Step Approach

To fully appreciate the formula, let's break it down into its key components:

  • 2Ï€y: This term represents the circumference of a circle formed by revolving a point (x, y) on the curve around the x-axis. As the curve revolves, each point traces out a circle, and this term calculates the length of that circle.
  • √(1 + (dy/dx)²): This is the arc length element, which approximates the length of a small segment of the curve. It is derived from the Pythagorean theorem and involves the derivative of the function, dy/dx, which represents the slope of the curve at a given point.
  • dx: This represents an infinitesimal change in x, the variable of integration. It indicates that we are summing up the areas of infinitesimally thin bands along the x-axis.
  • ∫[a, b]: This integral symbol signifies that we are summing up the areas of these infinitesimal bands over the interval [a, b], which defines the boundaries of the curve being revolved.

Applying the Formula to Our Specific Case: y = e(-x2)

Now that we have a solid understanding of the surface area formula, let's apply it to our specific problem: finding the surface area of the solid formed by revolving the curve y = e(-x2) over the interval [-3, 3] around the x-axis. This function, also known as the Gaussian function, plays a crucial role in various fields, including statistics, physics, and engineering. Its bell-shaped curve and rapid decay make it a fascinating subject for mathematical exploration.

Step 1: Finding the Derivative dy/dx

The first step in applying the surface area formula is to find the derivative of the function, dy/dx. This derivative represents the slope of the curve at any given point and is essential for calculating the arc length element. To find the derivative of y = e(-x2), we employ the chain rule, a fundamental concept in calculus that allows us to differentiate composite functions.

Applying the chain rule, we get:

dy/dx = e(-x2) * (-2x) = -2xe(-x2)

This derivative tells us how the function y changes with respect to x. It is a crucial piece of information for determining the shape and curvature of the function, which in turn affects the surface area of the solid formed by revolving it.

Step 2: Plugging into the Surface Area Formula

Now that we have the derivative, we can plug it into the surface area formula:

SA = 2π ∫[-3, 3] e(-x2) √(1 + (-2xe(-x2))²) dx

This expression represents the surface area of the solid formed by revolving the curve y = e(-x2) around the x-axis. It is an integral that needs to be evaluated to obtain the numerical value of the surface area. However, this integral does not have a simple closed-form solution, meaning we cannot find an elementary function that represents its antiderivative. Therefore, we must resort to numerical methods to approximate the value of the integral.

Step 3: Numerical Approximation

Since we cannot find an exact solution to the integral, we turn to numerical methods to approximate its value. Numerical integration techniques, such as the trapezoidal rule or Simpson's rule, provide accurate approximations of definite integrals by dividing the interval of integration into smaller subintervals and approximating the integral over each subinterval. In modern computational environments, we often rely on calculators or software with built-in numerical integration capabilities to solve such integrals efficiently.

Using a calculator or software with numerical integration capabilities, we can approximate the value of the integral:

SA ≈ 2π * 1.77245 ≈ 11.136

Step 4: Rounding to the Nearest Thousandth

The problem statement requests the surface area to the nearest thousandth. Therefore, we round our approximation to three decimal places:

SA ≈ 11.136

Conclusion: The Surface Area Revealed

Therefore, the surface area of the solid formed by revolving the function y = e(-x2) over the interval [-3, 3] around the x-axis is approximately 11.136 square units to the nearest thousandth. This result showcases the power of calculus in solving complex geometric problems, allowing us to determine the surface area of curved solids that would be impossible to calculate using traditional geometric methods.

This exploration has not only provided us with a numerical answer but also deepened our understanding of the concepts involved. We have journeyed through the surface area formula, dissected its components, and applied it to a specific function, all while emphasizing the importance of numerical methods in solving integrals that lack closed-form solutions. The world of calculus is filled with such intricate problems, and mastering these techniques empowers us to unravel the mysteries of mathematics and its applications in the real world.

The Significance of the Gaussian Function

It's worth noting the significance of the Gaussian function, y = e(-x2), in this context. This function, often referred to as the bell curve, is ubiquitous in various fields due to its unique properties. Its symmetry, smooth decay, and relationship to the normal distribution make it a cornerstone of statistics and probability theory. The fact that we can calculate the surface area of a solid formed by revolving this function around an axis further highlights its mathematical importance and its role in bridging different areas of mathematics.

In summary, the problem we tackled in this article demonstrates the power and elegance of calculus in solving real-world problems. By understanding the underlying concepts and utilizing the appropriate tools, we can unlock the secrets of complex geometric shapes and gain a deeper appreciation for the beauty and utility of mathematics. The surface area of revolution is just one example of the many fascinating applications of calculus, and this exploration serves as a stepping stone to further mathematical adventures.