Problem 50 - Thomas Calculus

Question

Sketching the change in a cube's volume The volume \(V=x^{3}\) of a cube with edges of length \(x\) increases by an amount \(\Delta V\) when \(x\) increases by an amount \(\Delta x\) . Show with a sketch how to represent \(\Delta V\) geometrically as the sum of the volumes of a. three slabs of dimensions \(x\) by \(x\) by \(\Delta x\) b. three slars of dimensions \(x\) by \(\Delta x\) by \(\Delta x\) c. one cube of dimensions \(\Delta x\) by \(\Delta x\) by \(\Delta x\) . The differential formula \(d V=3 x^{2} d x\) estimates the change in \(V\) with the three slabs.

Step-by-Step Solution

Answer

The change in volume \(\Delta V\) is a sum of three types of slabs and a small cube.

Step 1: Understanding the Volume Change

Initially, the volume of the cube is given by the formula \(V = x^3\), where \(x\) is the side length. When the side length increases by \(\Delta x\), the new volume becomes \((x + \Delta x)^3\). Hence, the change in volume, \(\Delta V\), can be represented by the expression \((x + \Delta x)^3 - x^3\).

Step 2: Expanding the New Volume

Expand \((x + \Delta x)^3\) using the binomial theorem: \[(x + \Delta x)^3 = x^3 + 3x^2 \Delta x + 3x (\Delta x)^2 + (\Delta x)^3\]Subtract the original volume \(x^3\) to find \(\Delta V\):\[\Delta V = 3x^2 \Delta x + 3x(\Delta x)^2 + (\Delta x)^3\]

Step 3: Geometric Interpretation of Components

Each term in the expression for \(\Delta V\) represents a different geometric volume:- **Three slabs of dimensions \(x \times x \times \Delta x\)**: This corresponds to the term \(3x^2 \Delta x\).- **Three slabs of dimensions \(x \times \Delta x \times \Delta x\)**: This corresponds to the term \(3x (\Delta x)^2\).- **One cube of dimensions \(\Delta x \times \Delta x \times \Delta x\)**: This is represented by \((\Delta x)^3\).

Step 4: Using the Differential Formula

The differential formula \(dV = 3x^2 dx\) is an approximation that sums up the volume changes represented by the three slabs \(3x^2 \Delta x\), ignoring the smaller second-order \(3x(\Delta x)^2\) and third-order \((\Delta x)^3\) components.

Key Concepts

Volume Change

In differential calculus, understanding how a function changes is crucial. Here, we're focusing on how the volume of a cube changes when its dimensions increase slightly. Initially, the volume of a cube is given by the formula \( V = x^3 \), where \( x \) represents the side length. When the side length increases by a small amount \( \Delta x \), the new volume becomes \((x + \Delta x)^3\). This results in a change in volume, denoted as \( \Delta V \). To compute \( \Delta V \), we subtract the original volume \( x^3 \) from the new volume \((x + \Delta x)^3\). This difference can be expressed in terms of \( x \) and \( \Delta x \), providing a detailed picture of how the volume changes due to the slight increase in edge length. This breakdown is a fundamental application of differential calculus, helping us understand the behavior of changing dimensions.

Cube Geometry

Cube geometry provides a tangible method to visualize and calculate changes in dimensions and their effect on volume. When a cube's edge length increases by a small value \( \Delta x \), the contributions to the change in volume \( \Delta V \) can be represented geometrically:

• **Three slabs with dimensions**: These slabs have dimensions \( x \times x \times \Delta x \) and correspond to the term \( 3x^2 \Delta x \). They represent the primary change in volume when the edge is extended in one direction.


• **Three slats with dimensions**: These slats add the next layer of volume change due to \( x \times \Delta x \times \Delta x \), represented by the term \( 3x(\Delta x)^2 \). They signify an expansion across two dimensions, adding a smaller contribution.


• **One small cube**: The final added volume comes from a tiny cube \( \Delta x \times \Delta x \times \Delta x \). This cube is represented by the term \((\Delta x)^3\), showing the volume added due to three-dimensional growth.

These components collectively give a clearer geometric perspective on how a small change in dimension impacts overall volume.

Binomial Expansion

To further understand the changes in the volume of a cube, binomial expansion becomes a useful tool. When calculating \( (x + \Delta x)^3 \), we use the binomial theorem:\[(x+\Delta x)^3 = x^3 + 3x^2\Delta x + 3x(\Delta x)^2 + (\Delta x)^3\]This expanded form clearly breaks down the changes into manageable pieces:

• **First term** \( x^3 \): It represents the original volume of the cube.


• **Second term** \( 3x^2\Delta x \): Represents the dominant part of the change in volume, corresponding to the primary slabs.


• **Third term** \( 3x(\Delta x)^2 \): A smaller change due to two-dimensional expansion.


• **Fourth term** \((\Delta x)^3 \) : The smallest, representing a new mini-cube created by growth in all dimensions.

The binomial expansion here offers more than just a mathematical depiction. It illustrates how even a small adjustment in size affects volume, showcasing the power of differential calculus in breaking down changes into understandable steps.