Problem 66 - Calculus of a Single Variable

Question

Approximating Arc Length or Surface Area In Exercises \(63-66,\) set up the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the are length or surface area. (You will learn how to evaluate this type of integral in Section \(8.8 . )\) Using a Loop Consider the graph of $$y^{2}=\frac{1}{12} x(4-x)^{2}$$ shown in the figure. Find the area of the surface formed when the loop of this graph is revolved about the \(x\) -axis.

Step-by-Step Solution

Answer

The exercise can be solved using the Formula for the Surface Area of a Volume of Revolution. An approximation of the Surface Area is obtained using a graphing utility. The exact answer depends on the accuracy of the available graphing utility.

Step 1: Re-arranging the equation

The first step is to re-arrange the equation into the form \(y = f(x)\). In this case, sqrt of both sides of the given equation which will result in \(y = \sqrt{\frac{1}{12} x(4-x)^{2}}\).

Step 2: Finding the derivative

The second step is Find the derivative of f(x) to include in the Surface Area Formula. The derivative of \(f(x) = \sqrt{\frac{1}{12} x(4-x)^{2}}\), may require chain rule, product rule and power rule for derivation.

Step 3: Applying the Formula for Surface Area

Having obtained \(y = f(x)\) and \(f'(x)\) the Surface Area can now be found. The limits for the definite integral must also be determined (starting and ending x-values of the loop). Once all the information is gathered, the Surface Area Formula \(2\pi \int_{a}^{b} f(x) \sqrt{1 + [f'(x)]^2} dx\), can be used to find the surface area.

Step 4: Use a Graphing Utility to Approximate the Result

The exercise suggests to approximate the definite integral using a graphing utility. Simply input the function from step 3 into the graphing utility and obtain the result. The result will be the surface area of the formed object when the loop of the given graph is revolved about the x-axis.

Key Concepts

Definite Integral

A definite integral is a mathematical expression used to calculate the area under a curve within specified bounds. It is represented by the integral symbol \( \int \) with limits of integration specified at the bottom and top (denoted as \( a \) and \( b \) respectively). When evaluating a definite integral, you find the total accumulation of quantities, such as area or arc length, between these limits.
For the problem at hand, the definite integral is used to compute the surface area of a solid of revolution, which means calculating the area of a surface formed by revolving a curve around an axis. The formula involves both the function \( f(x) \) and its derivative, indicative that understanding how the function changes is crucial.

• Make sure to accurately find the bounds for your specific problem, as they define the limits of integration.
• Utilize properties of integrals like linearity, if applicable, to simplify calculations.

By integrating the function defined as a specific mathematical expression, one can use numerical methods or graphing utilities to approximate these values when an analytical solution is difficult to obtain.

Revolution About the X-axis

Revolution about the x-axis refers to the method of creating a three-dimensional object by rotating a two-dimensional curve around the horizontal axis. This technique helps in visualizing and calculating volumes and surface areas of various geometrical shapes like cylinders and spheres. In this problem, the surface area is generated by rotating the loop of the curve given by \( y^2 = \frac{1}{12} x(4-x)^2 \) about the x-axis.
To set up the problem:

• Identifying the curve and changing it into \( y = f(x) \) form is essential for easier handling.
• Determine the limits of integration by finding the x-values where the curve intersects the x-axis, which are the same points where the loop starts and ends on the x-axis for revolution purposes.

By understanding the geometric manifestation of rotating curves around different axes, you gain a broader perspective of how calculus applies to real-world physics problems by extending 2D figures into 3D objects.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. A composite function is one in which a function is nested inside another, such as \( f(g(x)) \). The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function itself.
For the equation \( y = \sqrt{\frac{1}{12} x(4-x)^2} \), you need the chain rule to find the derivative efficiently. Breaking it down:

• The outer function here is a square root function, and the inner function involves \( x(4-x)^2 \).
• Apply the chain rule step-by-step: differentiate the outer function, then multiply by the derivative of the inner function.

Utilizing the chain rule in such problems is paramount because it simplifies daunting derivatives into manageable steps, allowing you to apply the surface area formula correctly with \( f'(x) \). Differentiation accuracy directly affects the calculation of physical quantities like surface area.

Graphing Utility

A graphing utility, such as a graphing calculator or software, is an invaluable tool for handling complex integrals and visualizing functions. These utilities are designed to perform numerical approximations, making them perfect for calculating definite integrals where analytical solutions are difficult or impossible.
When using a graphing utility to approximate the definite integral:

• Input the precise function \( f(x) \) and its derivative into the tool for accurate approximations.
• Ensure you define the correct integration bounds, as these affect the integral's result entirely.

Graphing utilities often provide graphical feedback, allowing you to observe the integration visually. This can enhance your understanding of concepts like the area under a curve or the surface of a revolution. As technology progresses, these tools become more sophisticated, offering greater precision and ease-of-use, proving indispensable in both educational and professional settings in solving calculus problems.