Problem 51 - Calculus

Question

Yogurt comes in a solid of revolution. Rotate the line \(y=m x\) around the \(y\) axis to find the volume between \(y=a\) and \(y=b\).

Step-by-Step Solution

Answer

The volume is \( V = \frac{\pi}{m} (b^2 - a^2) \).

Step 1: Understand the Problem

The problem asks us to find the volume of a solid of revolution by rotating the line \(y = mx\) about the \(y\)-axis between \(y = a\) and \(y = b\). This will require us to use the method of cylindrical shells for integration.

Step 2: Set Up the Integral

The formula for the volume of a solid of revolution using the method of cylindrical shells is \ The formula is: \[ V = 2\pi \int_{y=a}^{y=b} x \, dy \] Here, we rewrite \(x\) in terms of \(y\) using the line equation \(x = \frac{y}{m}\). Thus, our integral becomes: \[ V = 2\pi \int_{y=a}^{y=b} \frac{y}{m} \, dy \]

Step 3: Integrate to Find Volume

Calculate the integral: \[ V = 2\pi \int_{y=a}^{y=b} \frac{y}{m} \, dy = 2\pi \left[ \frac{1}{2m}y^2 \right]_a^b \] Simplify the expression:\[ V = \frac{\pi}{m} (b^2 - a^2) \]

Step 4: Write the Final Expression

Finally, substitute back into the simplified expression for the volume of the solid of revolution: \[ V = \frac{\pi}{m} (b^2 - a^2) \] This is the volume of the solid formed by revolving the line \(y=mx\) around the \(y\)-axis between \(y=a\) and \(y=b\).

Key Concepts

Method of Cylindrical Shells

The method of cylindrical shells is an important technique used to find volumes of solids that are created by rotating a region around a line, often one of the coordinate axes. By breaking down the solid into smaller pieces, the method calculates the volume by summing up the volumes of numerous hollow cylinders ("shells").

To use the cylindrical shells method, you need to:

• Select a point of rotation, often this is a line like the x-axis or y-axis.
• Split the solid into thin cylindrical shells using vertical or horizontal sections.
• Use the formula for finding the volume of each shell and integrate over the interval.

For the formula to calculate the total volume when rotating around the y-axis, you would use:\[ V = 2\pi \int_{y=a}^{y=b} x \ dy \]In this formula, \(x\) is the distance from the axis of rotation, and \(dy\) signifies a thin slice's height.

Volume Integration

Volume integration involves using integral calculus to determine the volume of a solid. In calculus, you can find the volume of a solid of revolution, such as in our exercise, by revolving a function around an axis. Using integration, it involves summing up an infinite number of infinitesimally small elements that make up the solid.

The integration process helps calculate the volume by:

• Setting up an integral that reflects the shape of the solid by considering slices or shells.
• Evaluating the integral across a defined interval that identifies the bounds, like between \(y = a\) and \(y = b\).
• Performing the integration to sum the infinitesimally small volume elements across the interval.

Utilizing volume integration, for the specific problem of rotating the line \(y=mx\) around the y-axis, we can transform the line into an x-expression by rearranging to get \(x= \frac{y}{m}\), and then integrate the function in terms of \(y\). This reflects how every part of mathematics synchronizes to solve problems involving volumes.

Rotating Curves

Rotating curves to form solids is a fascinating application of calculus concepts. By revolving a function or line around an axis, you can generate a three-dimensional solid that has depth and volume. This principle underlies many real-world applications such as in engineering and architecture.

Rotating a curve around an axis involves:

• Identifying the curve or line to be rotated, such as \(y=mx\) in this case.
• Determining the axis of rotation, here it is the y-axis.
• Using the method of cylindrical shells to find the volume by considering the distance each point on the curve is from the axis.

When a line is rotated, like in our problem scenario, we obtain a symmetric solid where its dimensions can be precisely calculated using calculus. This allows for the calculation of the volume of shapes that may be complex to handle using standard geometry alone. The ability to rotate curves and find volumes is a powerful tool in mathematical analysis.