Problem 196 - Calculus Volume 3

Question

Complete each task. The volume of a right circular cylinder is given by \(V(r, h)=\pi r^{2} h .\) Find the differential \(d V\). Interpret the formula geometrically.

Step-by-Step Solution

Answer

The differential is \(dV = (2\pi rh) dr + (\pi r^2) dh\). It expresses the change in volume due to changes in radius and height.

Step 1: Identify the variables and the functional relationship

The function given is the volume of a right circular cylinder, which depends on the radius \(r\) and the height \(h\). The function is given by \(V(r, h) = \pi r^2 h\). Here, \(V\) is the output, \(r\) and \(h\) are the inputs.

Step 2: Understand differentials

To find the differential \(dV\), understand that it represents a change in the volume when both \(r\) and \(h\) are slightly changed. The differential is calculated as a linear approximation of how \(V\) changes with small changes in \(r\) and \(h\).

Step 3: Apply the differential formula for \(V(r, h)\)

The differential \(dV\) can be found using the formula: \(dV = \frac{\partial V}{\partial r} dr + \frac{\partial V}{\partial h} dh\),where \(\frac{\partial V}{\partial r}\) and \(\frac{\partial V}{\partial h}\) are the partial derivatives of \(V\) with respect to \(r\) and \(h\), respectively.

Step 4: Calculate the partial derivatives

Calculate \(\frac{\partial V}{\partial r}\): Differentiate \(V(r, h) = \pi r^2 h\) with respect to \(r\) to get \(2\pi rh\). Calculate \(\frac{\partial V}{\partial h}\): Differentiate \(V(r, h) = \pi r^2 h\) with respect to \(h\) to get \(\pi r^2\).

Step 5: Substitute partial derivatives into the differential

Substitute the partial derivatives into the differential formula:\(dV = (2\pi rh) dr + (\pi r^2) dh\).

Step 6: Interpret the formula geometrically

Geometrically, the formula \(dV = (2\pi rh) dr + (\pi r^2) dh\) suggests how small changes in \(r\) and \(h\) affect the volume: - The term \((2\pi rh) dr\) represents the change in volume due to a change in the radius, indicating how much the volume increases for a small increase in \(r\) while keeping \(h\) constant.- The term \((\pi r^2) dh\) represents the change in volume due to a change in the height, indicating how much the volume increases for a small increase in \(h\) while keeping \(r\) constant.

Key Concepts

Partial Derivatives

Partial derivatives are an essential tool when dealing with functions of multiple variables. In the context of our problem, the volume of a cylinder, denoted as \(V(r, h) = \pi r^2 h\), depends on two variables: the radius \(r\) and the height \(h\).

When we talk about partial derivatives, we refer to the way a multivariable function changes as we tweak one input variable while keeping the others constant.

• If we compute the partial derivative with respect to \(r\), we focus on how the volume changes as only the radius changes, leading us to \(\frac{\partial V}{\partial r} = 2\pi rh\).
• On the other hand, computing it in relation to \(h\), where we only change the height, gives us \(\frac{\partial V}{\partial h} = \pi r^2\).

These partial derivatives help us express the total differential of the volume \(dV\), showing individual contributions from changes in radius and height.

Volume of a Cylinder

The volume of a cylinder is a fundamental concept in geometry and calculus, especially when dealing with real-world problems involving objects that have a circular base and a specific height.

A right circular cylinder's volume is calculated using the formula \(V = \pi r^2 h\), where:

• \(r\) is the radius of the cylinder's base.
• \(h\) is the height of the cylinder.
• \(\pi\approx 3.14159\) is the mathematical constant.

Understanding this formula is crucial as it becomes the basis for further calculations like finding differentials or analyzing how the volume changes with radius or height adjustments.

Geometric Interpretation of Differentials

The geometric interpretation of differentials allows us to visualize how small changes in a function's inputs affect its output. For the cylinder's volume, this can be visually represented by the formula \(dV = (2\pi rh) dr + (\pi r^2) dh\).

Here's what each part represents:

• \((2\pi rh) dr\) is the contribution to volume change from a small change in radius. Imagine slightly increasing the radius—this term tells us how much the cylinder's volume would grow as a result, using a cylindrical shell around the original cylinder.
• \((\pi r^2) dh\) shows the effect of changing the height. If the height increases slightly, this term shows the volume of the additional "disk" placed atop the original cylinder.

Through this formula, we see how small changes in geometry lead to changes in overall volume.

Multivariable Calculus

Multivariable calculus is the branch of calculus that deals with functions involving more than one variable. It's crucial for understanding how systems change with multiple influencing factors, like the radius \(r\) and height \(h\) in our cylinder volume problem.

This discipline uses concepts like partial derivatives and differentials to handle more complex scenarios than single-variable calculus can tackle. As seen in finding \(dV\),

• We combine partial derivatives to construct a differential that approximates changes in volume.
• The techniques of multivariable calculus allow us to solve practical problems in fields such as physics, engineering, and economics.

Understanding how multivariable calculus connects changes in one variable to the output of multivariable functions enhances our problem-solving toolkit significantly.