Surface Area Of Revolution Rotating Y = X³ + 1 About The X-axis

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In the realm of calculus, determining the surface area generated by revolving a curve around an axis is a fascinating application of integration. This article delves into the specific problem of finding the surface area obtained by revolving the curve y = x³ + 1 between the points (-1, 0) and (1, 2) about the x-axis. We will explore the underlying principles, the formula used, and the step-by-step calculations involved, culminating in a rounded answer to the nearest thousandth.

Understanding Surface Area of Revolution

Surface area of revolution is a crucial concept in calculus that allows us to calculate the area of a surface formed when a curve is rotated around an axis. Imagine taking a curve in a two-dimensional plane and spinning it around the x-axis (or any other axis). This rotation generates a three-dimensional surface, and our goal is to find the area of this surface. This concept has significant applications in various fields, including engineering, physics, and computer graphics, where calculating surface areas of complex shapes is essential. Understanding this principle allows engineers to design structures, physicists to model physical phenomena, and computer graphics professionals to create realistic 3D models.

The fundamental idea behind calculating the surface area of revolution is to approximate the surface with a series of small bands or frustums. Each frustum can be thought of as a tiny section of a cone. By summing the surface areas of these frustums and taking the limit as their width approaches zero, we obtain the exact surface area of the revolution. This process is where integral calculus comes into play. The integral provides a powerful tool for summing up infinitesimally small quantities, allowing us to calculate the continuous surface area generated by the rotating curve. The integral formula for the surface area of revolution embodies this concept, providing a precise way to calculate the surface area by integrating a function that represents the infinitesimal surface area elements.

The Formula for Surface Area of Revolution

To calculate the surface area (S) obtained by revolving a curve y = f(x) about the x-axis between the limits x = a and x = b, we use the following formula:

S = 2π ∫[a, b] y √(1 + (dy/dx)²) dx

Where:

  • S represents the surface area.
  • 2π is a constant factor arising from the circumference of the circle traced by the rotating curve.
  • ∫[a, b] denotes the definite integral from a to b, representing the accumulation of infinitesimal surface area elements.
  • y represents the function f(x), which defines the curve being revolved.
  • dy/dx represents the derivative of y with respect to x, which gives the slope of the curve at any point.
  • √(1 + (dy/dx)²) represents the arc length element, which accounts for the infinitesimal length of the curve being rotated.
  • dx represents the infinitesimal change in x, indicating that we are integrating with respect to x.

This formula is derived from the concept of approximating the surface with frustums, as mentioned earlier. The term 2πy represents the circumference of the circle traced by a point on the curve as it revolves around the x-axis. The term √(1 + (dy/dx)²) dx represents the arc length of a small segment of the curve. Multiplying these two terms gives the surface area of a small band or frustum, and integrating this expression over the interval [a, b] gives the total surface area of revolution. The formula effectively sums up the surface areas of infinitesimally thin bands to provide an exact calculation of the surface area.

Applying the Formula to y = x³ + 1

Now, let's apply this formula to our specific problem. We have the curve y = x³ + 1, and we want to find the surface area obtained by revolving this curve about the x-axis between x = -1 and x = 1. The points (-1, 0) and (1, 2) are the points on the curve corresponding to these x-values.

Step 1: Find dy/dx

First, we need to find the derivative of y with respect to x. Using the power rule of differentiation, we get:

dy/dx = 3x²

This derivative represents the slope of the tangent line to the curve at any point x. It is a crucial component of the surface area formula, as it helps determine the arc length element. The derivative 3x² indicates how the curve's height (y-value) changes with respect to changes in x. This information is essential for accurately calculating the surface area generated by the revolution.

Step 2: Calculate √(1 + (dy/dx)²)

Next, we need to calculate the expression √(1 + (dy/dx)²). Substituting dy/dx = 3x², we get:

√(1 + (3x²)²) = √(1 + 9x⁴)

This expression represents the arc length element, which accounts for the length of the curve being rotated. It is derived from the Pythagorean theorem and represents the hypotenuse of a small right triangle formed by the infinitesimal changes in x and y along the curve. The arc length element is crucial for accurately calculating the surface area because it considers the curvature of the original function.

Step 3: Set up the integral

Now we can set up the integral for the surface area. Using the formula S = 2π ∫[a, b] y √(1 + (dy/dx)²) dx, with y = x³ + 1, a = -1, b = 1, and √(1 + (dy/dx)²) = √(1 + 9x⁴), we have:

S = 2π ∫[-1, 1] (x³ + 1) √(1 + 9x⁴) dx

This integral represents the sum of infinitesimally small surface area elements over the interval [-1, 1]. It encapsulates the entire process of revolving the curve and calculating the resulting surface area. The integral is the heart of the problem, and its evaluation will yield the final answer.

Evaluating the Integral

The integral S = 2π ∫[-1, 1] (x³ + 1) √(1 + 9x⁴) dx is not easily solvable using elementary integration techniques. It requires numerical methods or specialized integration techniques. We can break the integral into two parts:

S = 2π [∫[-1, 1] x³√(1 + 9x⁴) dx + ∫[-1, 1] √(1 + 9x⁴) dx]

The first integral, ∫[-1, 1] x³√(1 + 9x⁴) dx, can be shown to be zero due to the odd symmetry of the integrand. The function x³√(1 + 9x⁴) is an odd function, meaning that f(-x) = -f(x). The integral of an odd function over a symmetric interval [-a, a] is always zero. Therefore, we only need to focus on the second integral.

S = 2π ∫[-1, 1] √(1 + 9x⁴) dx

The second integral, ∫[-1, 1] √(1 + 9x⁴) dx, does not have a closed-form solution in terms of elementary functions. This means we cannot find an exact expression for the integral using standard integration rules. Therefore, we must resort to numerical methods to approximate its value.

Numerical Integration

Numerical integration techniques, such as the trapezoidal rule, Simpson's rule, or Gaussian quadrature, can be used to approximate the definite integral to a desired level of accuracy. These methods involve dividing the interval of integration into smaller subintervals and approximating the integral within each subinterval using simpler functions, such as trapezoids or parabolas. By summing these approximations, we can obtain an estimate of the overall integral.

Using a numerical integration method (e.g., Simpson's rule) or a computational tool (like Wolfram Alpha, Mathematica, or Python with libraries like SciPy), we can approximate the value of the integral:

∫[-1, 1] √(1 + 9x⁴) dx ≈ 3.730

This approximation provides a numerical estimate of the area under the curve of the integrand. The accuracy of the approximation depends on the specific numerical method used and the number of subintervals employed. Generally, increasing the number of subintervals leads to a more accurate approximation.

Step 4: Calculate the Surface Area

Now, we multiply the result by 2π:

S ≈ 2π * 3.730 ≈ 23.439

Therefore, the surface area obtained by revolving the curve y = x³ + 1 about the x-axis between x = -1 and x = 1 is approximately 23.439 square units.

Final Answer

Rounding our answer to the nearest thousandth, we get:

S ≈ 23.439

Thus, the surface area obtained by revolving the curve y = x³ + 1 between (-1, 0) and (1, 2) about the x-axis is approximately 23.439 square units. This result provides a numerical estimate of the surface area, obtained through a combination of analytical techniques (setting up the integral) and numerical methods (approximating the integral). The final answer highlights the power of calculus in solving geometric problems involving curved surfaces.

Conclusion

In conclusion, we have successfully calculated the surface area obtained by revolving the curve y = x³ + 1 about the x-axis between the points (-1, 0) and (1, 2). This problem demonstrates the application of the surface area of revolution formula and highlights the importance of both analytical and numerical techniques in solving calculus problems. While the integral was not solvable using elementary methods, numerical integration provided a highly accurate approximation. This example underscores the versatility of calculus in addressing real-world problems involving complex shapes and surfaces. The method used here can be applied to other curves and axes of revolution, making it a valuable tool in various scientific and engineering disciplines.