Problem 38 - Thomas Calculus

Question

The region in the first quadrant that is bounded above by the curve \(y=1 / \sqrt{x},\) on the left by the line \(x=1 / 4,\) and below by the line \(y=1\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid by a. the washer method. \(\quad\) b. the shell method.

Step-by-Step Solution

Answer

The volume of the solid is \( \frac{5\pi}{96} \) using both methods.

Step 1: Identifying the Region

We need to understand the region of integration. The region is bounded above by \( y = \frac{1}{\sqrt{x}} \), left by the line \( x = \frac{1}{4} \), and below by the line \( y = 1 \). This region is in the first quadrant.

Step 2: Setup for Washer Method

For the washer method, we integrate with respect to \( y \). We need to express \( x \) in terms of \( y \), which from \( y = \frac{1}{\sqrt{x}} \), gives us \( x = \frac{1}{y^2} \). The bounds for integration are \( y = 1 \) to \( y = 2 \) since the curve \( y = \frac{1}{\sqrt{x}} \) intersects the \( x = \frac{1}{4} \) at \( y = 2 \).

Step 3: Volume with Washer Method

The volume is given by the formula for the washer method: \[ V = \pi \int_{y_1}^{y_2} \left((\text{outer radius})^2 - (\text{inner radius})^2 \right) \, dy \]Here, outer radius \( R(y) = \frac{1}{y^2} \) and inner radius is the distance from \( y \)-axis to \( x = \frac{1}{4} \) which is a constant \( \frac{1}{4} \). The integral becomes:\[ V = \pi \int_{1}^{2} \left( \left(\frac{1}{y^2}\right)^2 - \left(\frac{1}{4}\right)^2 \right) \, dy \]

Step 4: Evaluation of Volume with Washer Method

Evaluating the integral:\[ V = \pi \int_{1}^{2} \left(\frac{1}{y^4} - \frac{1}{16}\right) \, dy \]\[ V = \pi \left[ \int_{1}^{2} \frac{1}{y^4} \, dy - \int_{1}^{2} \frac{1}{16} \, dy \right] \]\[ V = \pi \left[ \left( -\frac{1}{3y^3} \right) \Big|_{1}^{2} - \frac{1}{16} (y) \Big|_{1}^{2} \right] \]\[ V = \pi \left[\left(-\frac{1}{24} + \frac{1}{3}\right) - \frac{1}{16}(1) \right] \]\[ V = \pi \left[ \frac{7}{24} - \frac{1}{16} \right] \]Simplifying, we find:\[ V = \frac{5\pi}{96} \]

Step 5: Setup for Shell Method

For the shell method, we integrate with respect to \( x \). The bounds for integration are from \( x = \frac{1}{4} \) to where \( y = 1 \), which is \( x = 1 \). The height for each shell is \( \frac{1}{\sqrt{x}} - 1 \), and the radius for each shell is \( x \).

Step 6: Volume with Shell Method

The volume is given by the formula for the shell method: \[ V = 2\pi \int_{x_1}^{x_2} (\text{radius})(\text{height}) \, dx \]The integral becomes:\[ V = 2\pi \int_{1/4}^{1} x \left( \frac{1}{\sqrt{x}} - 1 \right) \, dx \]

Step 7: Evaluation of Volume with Shell Method

The integral simplifies to:\[ V = 2\pi \left( \int_{1/4}^{1} x^{1/2} \, dx - \int_{1/4}^{1} x \, dx \right) \]Computing each term:\[ V = 2\pi \left[ \frac{2}{3}x^{3/2} \Big|_{1/4}^{1} - \frac{1}{2}x^2 \Big|_{1/4}^{1} \right] \]\[ V = 2\pi \left[ \frac{2}{3} \left(1 - \frac{1}{8}\right) - \frac{1}{2} \left(1 - \frac{1}{16}\right) \right] \]\[ V = 2\pi \left[ \frac{14}{24} - \frac{15}{32} \right] \]Simplifying, we find:\[ V = \frac{5\pi}{96} \]

Step 8: Comparison of Methods

Both the washer method and the shell method yield the same volume for the solid, which confirms the accuracy of both methods.

Key Concepts

Understanding Volume of Solids of Revolution

Understanding the volume of solids of revolution is essential when dealing with calculus-related integration problems. A solid of revolution is generated by rotating a two-dimensional region around a specified axis. This rotation forms a three-dimensional object whose volume can be calculated.
The two primary methods for finding the volume of these solids are the washer method and the shell method. Each technique accounts for the solid's unique characteristics and revolving nature.
Essentially,

• The washer method involves integrating across the solid's cross-sectional area, shaped like washers (or disks with holes), perpendicular to the axis of revolution.
• The shell method, on the other hand, deals with cylindrical shells formed parallel to the axis of rotation.

Both approaches involve slicing the solid in a way that simplifies the integration calculations.

Mastering Integration Techniques

Integration techniques are crucial when solving problems related to the volume of solids of revolution. These methods involve calculating integrals where derivatives are not evidently reversible. There are specific integration techniques tailored for use with the washer and shell methods.
In the washer method, the integration is often with respect to the variable perpendicular to the axis of rotation. This requires setting up and solving integrals that measure the volume removed by the hole in each washer.
In contrast, the shell method utilizes integration along the axis of rotation, focusing on summing up cylindrical shells' volumes. Each shell's radius is the distance from the curve to the axis, and its height is determined by subtracting one function from another.

• For the washer method: Proper understanding of the bounding curves and limits is crucial, requiring careful treatment of the outer and inner radii.
• For the shell method: The setup usually involves recognizing how to express the problem's components, such as height and radius, before integrating.

Being adept at these techniques allows for precise volume calculation.

Applications of Calculus in Real-World Problems

Calculus is not only theoretical but also deeply practical in various applications across multiple fields. Determining the volume of solids of revolution epitomizes the thrilling intersection between abstract mathematics and real-world problems. Such calculations are critical in physics, engineering, and medical fields, to name a few.
In engineering, understanding these volumes helps in designing components like pressure vessels and tanks, where knowing precise volumes is critical. Similarly, calculus applies to the medical field in simulation and modeling of organs, where accurate volume measurements can lead to better diagnosis and treatments.

• In manufacturing: Calculus assists in optimizing shapes and components for efficiency and material usage.
• In architecture: Solids of revolution and integration techniques help design structures that are both functional and aesthetically pleasing.

Overall, mastering calculus concepts offers powerful tools to solve complex real-world challenges effectively.