Problem 279 - Calculus Volume 3

Question

For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface. $$ \begin{aligned} &\begin{array}{llll} 279 . & {[\mathrm{T}]} & \text { Half } & \text { cylinder } \end{array}\\\ &\\{(r, \theta, z): r=4,0 \leq \theta \leq \pi, 0 \leq z \leq 7\\} \end{aligned} $$

Step-by-Step Solution

Answer

The approximate surface area of the half-cylinder is \(87.96\) square units.

Step 1: Understand the Parametric Equation of the Surface

The given surface is a half-cylinder described by the parametric equations: \( r = 4 \), where \( 0 \leq \theta \leq \pi \) and \( 0 \leq z \leq 7 \). This represents a cylinder, with radius 4, that spans half a circle and extends along the z-axis from 0 to 7.

Step 2: Parameterize the Surface

The parametric equations for the half-cylinder surface are given by:\[x(\theta, z) = 4 \cdot \cos(\theta),y(\theta, z) = 4 \cdot \sin(\theta),z(\theta, z) = z\]where \(0 \leq \theta \leq \pi\) and \(0 \leq z \leq 7\).

Step 3: Calculate the Surface Area with a Computer Algebra System

To calculate the surface area, we use the formula for the surface area of a parametric surface:\[A = \int_{a}^{b} \int_{c}^{d} \left\| \frac{\partial \mathbf{r}}{\partial \theta} \times \frac{\partial \mathbf{r}}{\partial z} \right\| \, d\theta \, dz\]where \( \mathbf{r}(\theta, z) = (x(\theta, z), y(\theta, z), z(\theta, z)) \). Calculate the partial derivatives, cross product, norm, and integrate over the specified bounds.

Step 4: Compute Partial Derivatives

Compute the partial derivatives of the parametric equations:\( \frac{\partial \mathbf{r}}{\partial \theta} = (-4 \cdot \sin(\theta), 4 \cdot \cos(\theta), 0) \) and\( \frac{\partial \mathbf{r}}{\partial z} = (0, 0, 1) \).

Step 5: Cross Product of Partial Derivatives

Calculate the cross product of \( \frac{\partial \mathbf{r}}{\partial \theta} \) and \( \frac{\partial \mathbf{r}}{\partial z} \):\[ \frac{\partial \mathbf{r}}{\partial \theta} \times \frac{\partial \mathbf{r}}{\partial z} = (4 \cdot \cos(\theta), 4 \cdot \sin(\theta), 0) \]

Step 6: Magnitude of the Cross Product

Find the magnitude of the cross product vector:\[ \left\| (4 \cdot \cos(\theta), 4 \cdot \sin(\theta), 0) \right\| = \sqrt{(4 \cdot \cos(\theta))^2 + (4 \cdot \sin(\theta))^2} = 4 \]

Step 7: Integrate to Find Surface Area

Substitute the magnitude into the surface area integral:\[ A = \int_{0}^{\pi} \int_{0}^{7} 4 \, dz \, d\theta = 4 \times \int_{0}^{\pi} \int_{0}^{7} 1 \, dz \, d\theta \]Compute the integral: \( 4 \times \left[ 7 \times (\pi - 0) \right] = 28\pi \).

Step 8: Calculate and Approximate the Final Area

Multiply the result to approximate the numerical value of the surface area:\( 28\pi \approx 87.96 \).

Key Concepts

Parametric Equations

Parametric equations are a fundamental concept in studying surfaces and curves in space. They express the coordinates of the points on a surface or curve as functions of one or more parameters. In the case of our half-cylinder, the surface is defined by the parameters \( \theta \) and \( z \). This approach allows us to define complex surfaces and analyze their properties systematically.

For a half-cylinder, the parametric equations are given by:

• \( x(\theta, z) = 4 \cdot \cos(\theta) \)
• \( y(\theta, z) = 4 \cdot \sin(\theta) \)
• \( z(\theta, z) = z \)

These equations effectively plot every point on the surface where the radius is 4 across a semi-circular path, and the height ranges from 0 to 7 units along the z-axis.

Understanding parametric equations allows us to plot these surfaces on graphing software or mathematics tools, enabling further calculations such as surface area or volume.

Computer Algebra System

A Computer Algebra System (CAS) is software designed to execute symbolic mathematical computations. Such systems are capable of performing algebraic operations, differentiating symbols, integrating functions, and even solving equations, all algebraically.

Using a CAS in our task allows us to handle complex calculations and integrations that could be labor-intensive if done manually. These tools can efficiently carry out procedures, such as finding the cross product of derivatives and determining integrals to approximate a surface area, like in our half-cylinder.

Some popular CAS software includes:

• Mathematica
• Maple
• SymPy in Python

By utilizing these systems, students can explore mathematical concepts more deeply, checking their work for accuracy while learning the necessary steps for solving similar problems independently.

Cross Product

The cross product is a vector operation used extensively in three-dimensional space mathematics. It involves two vectors and results in another vector that is perpendicular to the plane containing the original vectors. In our surface area calculation, we calculate the cross product of partial derivatives of the parametric equations to find the vector representing a section of the surface.

The cross product for

• \( \frac{\partial \mathbf{r}}{\partial \theta} = (-4 \cdot \sin(\theta), 4 \cdot \cos(\theta), 0) \)
• \( \frac{\partial \mathbf{r}}{\partial z} = (0, 0, 1) \)

results in \( \frac{\partial \mathbf{r}}{\partial \theta} \times \frac{\partial \mathbf{r}}{\partial z} = (4 \cdot \cos(\theta), 4 \cdot \sin(\theta), 0) \). This vector is crucial in calculating the surface area because it denotes a differential element of the surface.

The direction and magnitude of the cross product vector help us understand the orientation and size of these surface elements.

Integral Calculus

Integral calculus is a key branch of calculus focused on accumulation quantities, such as areas under curves or volumes—vital in calculating the surface area of three-dimensional shapes. In this exercise, integral calculus helps us sum up the surface elements to find the total area of the half-cylinder.

By integrating the magnitude of the cross product over the specified bounds, we calculate the total surface area:\[A = \int_{0}^{\pi} \int_{0}^{7} 4 \, dz \, d\theta = 4 \times \int_{0}^{\pi} \int_{0}^{7} 1 \, dz \, d\theta \]

This integral accumulates the contribution of each tiny piece of surface described by our parametric definitions over the whole half-cylinder. Therefore, integral calculus transforms our 'patches' of surface area into a comprehensive total area—calculated as \( 28\pi \approx 87.96 \) after performing the multiplication post integration.