Problem 15 - Basic Technical Mathematics with Calculus

Question

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case. $$y=\sqrt{x}, y=1, y=2, x=0 \quad \text { (shells) }$$

Step-by-Step Solution

Answer

The volume is \( \frac{15\pi}{2} \).

Step 1: Understand the Problem

We need to find the volume of the solid formed by revolving the area bounded by the curves \( y = \sqrt{x} \), \( y = 1 \), \( y = 2 \), and \( x = 0 \) around the x-axis using the method of cylindrical shells.

Step 2: Set up the Integral for Shell Method

The shell method formula for volume is given by \( V = \int_{a}^{b} 2\pi (radius)(height) \, dx \). Here the radius is \( y \) and the height of the shell is the extent of \( x \) as a function of \( y \), i.e., \( x = y^2 \). The bounds for \( y \) are from \( y = 1 \) to \( y = 2 \).

Step 3: Substitute in the Shell Method Formula

Substitute \( x = y^2 \), radius = \( y \), height = \( y^2 \), and the bounds into the formula to get: \[ V = \int_{1}^{2} 2\pi (y)(y^2) \, dy = \int_{1}^{2} 2\pi y^3 \, dy. \]

Step 4: Evaluate the Integral

Evaluate the integral: \[ V = 2\pi \int_{1}^{2} y^3 \, dy = 2\pi \left[ \frac{y^4}{4} \right]_{1}^{2}. \]

Step 5: Calculate the Result

Plug in the upper and lower limits into the evaluated integral: \[ V = 2\pi \left( \frac{2^4}{4} - \frac{1^4}{4} \right) = 2\pi \left( \frac{16}{4} - \frac{1}{4} \right) = 2\pi \cdot \frac{15}{4}. \] The result simplifies to \[ \frac{15\pi}{2}. \]

Key Concepts

Volume of Revolution

The concept of volume of revolution involves creating a 3-dimensional solid by rotating a 2-dimensional region around a specified axis.
This is an intriguing application in calculus that provides deeper insight into how shapes behave in three-dimensional space.
To find this volume, certain integration methods are used to calculate all the infinitesimal slices of the solid and aggregate them.

• **Basic idea**: Imagine taking a flat sheet, defined by some curves, then rotating it around an axis. As each slice rotates, it forms a new part of the volume.
• **Importance**: This concept is crucial in engineering and sciences where rotational bodies like tanks and gears need precise volume calculations.

In calculus, your choice of axis and method (like the cylindrical shell method or the disk/washer method) determines how you set up the integrals. This introduces flexibility in solving a problem based on how complex the region or functions are. Understanding this concept is essential for solving various practical and theoretical problems related to volumes and shapes.

Cylindrical Shell Method

The cylindrical shell method is a powerful technique to find the volume of a solid of revolution when the solid is rotated around an axis, usually different from the axis parallel to which the function runs.
This method can often simplify calculations that would be cumbersome with other methods.

• **Key Idea**: Revolve thin strips around an axis to form cylindrical "shells". By summing up the volumes of infinitesimally thin shells, we find the total volume of the solid.
• **When to Use**: Ideal when functions are easier to integrate in terms of the other variable. In our example, revolving around the x-axis, but functions naturally describe a slice in terms of y.

The cylindrical shell method comes into play when setting up the integral as:\[V = \int_{a}^{b} 2\pi (radius)(height) \, dx\]
Where the 'radius' typically represents the given variable, while the 'height' is determined by expressions involving the other variable. Substituting these into the formula constructs the integral, simplifying our path to solving for the volume.

Definite Integral

A definite integral is the principal tool in calculus for computing the accumulation of quantities. In the context of volume of revolution problems, it helps calculate the total volume formed by adding up all the small parts.
This method gives precise numerical results that are crucial in practical applications.

• **Core Function**: Represents the limit of a sum of areas or, in our case, volumes, which helps in measuring space within defined bounds.
• **Applications**: Used extensively in areas like physics, engineering, and economics to find quantities accumulated over time or space.

The definite integral uses limits of integration—upper and lower bounds—to ensure accurate calculations over the specific region of interest. In our exercise, we set up the definite integral from 1 to 2 with respect to y to find out the specified volume, ensuring the outcome is both exact and proper.

Integration Techniques

In calculus, integration techniques are strategies used to solve integrals that aren’t straightforward, often making complex problems tractable.
These techniques are essential for solving problems related to the volume of revolutions as well as a myriad of other mathematical and applied models.

• **Common Techniques**: Methods like substitution, integration by parts, partial fraction decomposition, etc., allow for the simplification of complex expressions before integration.
• **Implementation**: Determining an appropriate method ensures the integral can be solved analytically or, when necessary, via numerical methods.

In our example, the integration is direct as the function is straightforward, but for more complex shapes, mastering various techniques becomes invaluable. Calculus problems often involve choosing the most efficient technique to find solutions quickly and accurately.