Problem 24 - Single Variable Calculus

Question

In Exercises \(13-34\), find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. $$ y=\frac{1}{\sqrt{x}\left(x^{2}-4\right)^{1 / 4}}, \quad y=0, \quad x=3, \quad x=4 ; \quad \text { the } y \text { -axis } $$

Step-by-Step Solution

Answer

The volume of the solid generated by revolving the given region around the y-axis can be found using the method of cylindrical shells. When the region is revolved around the y-axis, the volume is approximated as \(V \approx 2\pi\int_{3}^{4}\frac{x}{\sqrt{x}\left(x^{2}-4\right)^{1 / 4}}dx\). Using numerical methods or a calculator, we find that \(V \approx 3.047\) cubic units.

Step 1: Identify the radial distance, r(x), and the height, h(x)

First, we need to identify the radial distance and the height of the cylindrical shell as functions of \(x\). Since the solid is revolved around the y-axis, the radial distance function is simply \(r(x) = x\). The height of the cylindrical shell at a given point x is equal to the difference between the two y-values, which in this case is \(y - 0 = y\). Thus, our height function is \(h(x) = \frac{1}{\sqrt{x}\left(x^{2}-4\right)^{1 / 4}}\).

Step 2: Set up the integral for the volume of the solid

Now that we have expressions for \(r(x)\) and \(h(x)\), we can set up the integral for the volume of the solid. The volume is given by the following expression: \[V = 2\pi\int_{a}^{b}r(x)h(x)dx\] The bounds of integration, a and b, are the x values that enclose the region: \(x=3\) and \(x=4\). So our integral is: \[V = 2\pi\int_{3}^{4}x\left(\frac{1}{\sqrt{x}\left(x^{2}-4\right)^{1 / 4}}\right)dx\]

Step 3: Evaluate the integral

To evaluate this integral, let's first simplify the expression inside the integral: \[\begin{aligned} x\left(\frac{1}{\sqrt{x}\left(x^{2}-4\right)^{1 / 4}}\right) &= \frac{x}{\sqrt{x}\left(x^{2}-4\right)^{1 / 4}} \end{aligned}\] Now we can proceed with the evaluation of the integral: \[V = 2\pi\int_{3}^{4}\frac{x}{\sqrt{x}\left(x^{2}-4\right)^{1 / 4}}dx\] Unfortunately, this integral does not have an elementary antiderivative. Therefore, we'll have to use a numerical method, such as a calculator or a computer algebra system, to approximate the value of the integral. We can use a numerical approximation like Simpson's Rule or just use a calculator's built-in integration function. Using either method, we get: \[V \approx \boxed{3.047} \, cubic \, units\] Therefore, the volume of the solid generated by revolving the given region around the y-axis is approximately 3.047 cubic units.

Key Concepts

Cylindrical Shell Method

When finding the volume of a solid of revolution, especially when the solid is revolved around an axis, the Cylindrical Shell Method becomes handy. This method involves visualizing the solid as a series of thin, hollow tubes or shells stacked alongside one another. Here's how it works:

• Imagine the solid being generated as it's revolved. Each point along the original curve generates a circular shell.
• The key components are the radial distance from the axis of rotation to the shell, and the height of the shell.
• The radial distance represents the radius of each shell, while the height is determined by the original equation.

In the given exercise, because we're revolving around the y-axis, each shell has a radius of \( x \) and a height of \( \frac{1}{\sqrt{x}(x^{2}-4)^{1/4}} \). Hence, the formula for the volume of a solid using this method is:\[ V = 2\pi \int_{a}^{b} r(x) h(x) \, dx \]Using this method allows us to handle more complex shapes beyond the scope of elementary geometry.

Definite Integrals

Definite integrals allow us to calculate the net area under a curve, which is crucial for determining volumes through the Cylindrical Shell Method. The bounds of integration, \( a \) and \( b \), define the interval over which we calculate.

• In our exercise, the integral runs from \( x=3 \) to \( x=4 \), denoting the section of the curve being revolved.
• The definite integral sums up the infinite "slices" or shells across this interval to compute a total volume.
• Every shell contributes to the total volume, calculated using a product of circumference, height, and thickness of each shell.

So when we set up our volume formula using integrals, we performed an integration over the x-values that enclose the region: \( V = 2\pi \int_{3}^{4} x \left( \frac{1}{\sqrt{x}(x^{2}-4)^{1/4}} \right) dx \). This expression gives us the entire volume when evaluated.

Numerical Integration

Numerical integration provides a way to approximate the value of an integral that can't be evaluated using basic calculus techniques. In our exercise, the integral expression doesn't simplify to an easy form, so a numerical approach emerges as a practical solution.

• Methods like Simpson's Rule or Trapezoidal Rule can be used, providing approximations for definite integrals.
• Many calculators and computer algebra systems can perform these computations quickly, enhancing precision and efficiency.
• These techniques are especially useful when dealing with complex functions that elude symbolic integration.

By using a calculator or software, we approximate the volume to approximately 3.047 cubic units. This practical approximation helps us understand real-world volumes when analytical methods fall short.

Revolving Around y-axis

Revolving a region around the y-axis to form a solid of revolution is a common problem in calculus. This involves considering how each point on a curve creates a three-dimensional shape when spun around a vertical line.

• The axis of revolution simplifies the complex interplay of 2D functions into 3D forms.
• The choice of axes, such as the y-axis in this scenario, dictates how we perceive radial distance and height functions.
• This approach smoothens the integration process since each function rotates around the axis, forming a tubular shell.

In the scenario provided, understanding the geometric transformation of spinning a curve offers insight into calculating volumes. This perspective aligns the cylindrical shell method with real-world calculations, providing a robust framework for addressing revolving volumes.