Problem 9 - Calculus Single Variable

Question

In each of Exercises 7-12, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region to the right of the \(y\) -axis, to the left of the curve \(y=\ln (x)\), above the \(x\) -axis, and below \(y=1\).

Step-by-Step Solution

Answer

The volume is \( \frac{\pi}{2} (e^2 - 1) \).

Step 1: Visualize the Region and Solid

First, let's understand the region \( \mathcal{R} \). The region is bounded by the \( y \)-axis on the left, \( y = \ln(x) \) on the right, the \( x \)-axis below, and the line \( y = 1 \) above. We are rotating this region about the \( y \)-axis to form a solid.

Step 2: Set Up the Integral Using the Disk Method

Since we're rotating about the \( y \)-axis, we express \( x \) as a function of \( y \). From \( y = \ln(x) \), we have \( x = e^y \). The radius of a representative disk, when sliced perpendicular to the \( y \)-axis, is the distance from the \( y \)-axis to the curve, which is \( x = e^y \). The thickness of each disk is \( dy \).

Step 3: Determine the Limits of Integration

The limits of integration are determined by the \( y \)-coordinates of \( \mathcal{R} \). From the problem statement, \( \mathcal{R} \) lies above the \( x \)-axis (\( y = 0 \)) and below \( y = 1 \). Therefore, the limits of integration are from \( y = 0 \) to \( y = 1 \).

Step 4: Write the Volume Integral

The volume \( V \) of the solid is given by the integral: \[ V = \pi \int_{0}^{1} (e^y)^2 \, dy \] This is due to the fact that the area of a disk with radius \( e^y \) is \( \pi (e^y)^2 \).

Step 5: Solve the Integral

To find the volume, evaluate the integral: \[ V = \pi \int_{0}^{1} e^{2y} \, dy \] The antiderivative of \( e^{2y} \) is \( \frac{1}{2}e^{2y} \). Thus, \[ V = \pi \left[ \frac{1}{2} e^{2y} \right]_{0}^{1} = \pi \left( \frac{1}{2} e^2 - \frac{1}{2} e^0 \right) \] This simplifies to \[ V = \pi \left( \frac{1}{2} e^2 - \frac{1}{2} \right) = \frac{\pi}{2} (e^2 - 1) \].

Step 6: Presenting the Final Volume

The volume of the solid obtained by rotating the region \( \mathcal{R} \) about the \( y \)-axis is \( \frac{\pi}{2} (e^2 - 1) \).

Key Concepts

Volume of Solids

The volume of a solid is the measure of three-dimensional space that it occupies. Calculating this volume becomes essential when dealing with shapes or regions that are rotated around a specific axis—producing solids of revolution. By using the disk method, one can easily compute these volumes by considering the rotated area as composed of several thin circular disks.

Here is how it works:

• Envision the area or region you want to rotate and understand its boundaries.
• Imagine slicing the solid into thin disks perpendicular to the axis of rotation.
• Each disk forms a tiny cylinder where the thickness is infinitely small denoted by the differential in the axis.

This method involves setting up an integral that adds up all the volumes of these individual disks. By integrating the area of each disk over the region's boundaries, you can determine the solid's total volume.

Integration

Integration is a mathematical technique used to calculate several quantities, including the area under a curve and, importantly, in this context, the volume of solids. When dealing with volumes of solids of revolution, integration helps accumulate the volume of infinitesimally small slices to find the total volume of the solid.

The process involves setting up an integral that represents the sum of these tiny volumes. In the case of the disk method, you calculate the volume of each slice having thickness dy or dx, depending on the axis of rotation.

Here's a clearer look at how it works:

• Express the function that defines the boundary of your region in terms of the variable you are integrating against.
• Set the limits of integration to match where the region starts and ends with respect to the axis.
• Apply the formula for the disk method, and compute the integral. In our case, the integral \[ V = \pi \int_{y_{1}}^{y_{2}} (f(y))^2 \, dy \]

Completing this integration provides the volume of the solid of revolution.

Rotational Symmetry

Rotational symmetry in mathematics refers to a figure or shape that looks the same even after being rotated about an axis. When a two-dimensional region is rotated around an axis, it generates a three-dimensional solid that possesses rotational symmetry.

This symmetry is fundamental when you're calculating the volume of a solid of revolution. For example, a circular disk possesses rotational symmetry around its center. When such regions are rotated around an axis, the resulting solid has a consistent and symmetrical shape.

This concept simplifies computations since it ensures uniformity throughout the shape. While integrating using the disk method:

• The axis of rotation is crucial. It determines the direction and nature of the disks' arrangement.
• The resultant solid maintains symmetry, which is vital for ensuring all slices or disks are identical vertically.

This rotational arrangement underpins the consistent application of the disk method formula, ensuring accuracy in volume calculations.