Problem 30 - Calculus Single Variable

Question

In each of Exercises 25-30, use the method of cylindrical shells 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 below the graph of \(y=2 / x, 1 \leq x \leq 2\) and above the \(x\) -axis.

Step-by-Step Solution

Answer

The volume is \(4\pi\).

Step 1: Understand the Problem

We are asked to find the volume of a solid formed by rotating the region \( \mathcal{R} \) below the curve \( y = \frac{2}{x} \) for \( 1 \leq x \leq 2 \) and above the \( x \)-axis, about the \( y \)-axis.

Step 2: Setup the Formula for Cylindrical Shells

In cylindrical shells method, the volume \( V \) is calculated as:\[V = \int_{a}^{b} 2\pi \cdot \text{radius} \cdot \text{height}\, dx\]Here, the radius is \( x \) and the height is determined by the function \( y = \frac{2}{x} \). Therefore, the volume is given by:\[V = \int_{1}^{2} 2\pi \cdot x \cdot \frac{2}{x}\, dx\]

Step 3: Simplify and Set Up the Integral

Simplify the expression inside the integral:\[2\pi \cdot x \cdot \frac{2}{x} = 4\pi\]Then, the integral becomes:\[V = \int_{1}^{2} 4\pi \, dx\]

Step 4: Integrate

Now, integrate the expression: \[V = 4\pi \int_{1}^{2} \, dx = 4\pi \left[ x \right]_{1}^{2}\]Evaluate this integral:\[4\pi \left[ x \right]_{1}^{2} = 4\pi (2 - 1) = 4\pi \cdot 1 = 4\pi\]Thus, the volume \( V \) is \( 4\pi \).

Step 5: Final Statement

The volume \( V \) of the solid obtained by rotating the region \( \mathcal{R} \) about the \( y \)-axis is \( 4\pi \).

Key Concepts

Volume of Revolution

When pondering the concept of a volume of revolution, picture transforming a flat region into a three-dimensional shape by spinning it around an axis.
This technique helps you find the volume of a solid created from rotating a region around a line, often the x-axis or y-axis.
The region in focus here is a slice of the plane that turns into a 3D solid when rotated.

• Consider a region under a curve and above the x-axis.
• Imagine this region taking a full spin around an axis, crafting a spatial figure akin to a vase or a cone.
• This transformation yields what's called a 'rotational solid.'

Thus, calculating the volume of revolution requires understanding how a flat shape evolves into a full-bodied 3D solid.
It's like discovering the mass of a clay pot by understanding the dimensions of its raw, unfired slab.

Integral Calculus

Integral calculus delves into the accumulation of quantities, such as areas under curves or the volumes of solids.
Primarily, it's about piecing together small sections to form a whole.
In this specific problem, we're examining how segments of the region under the curve accumulate into the volume of the 3D solid.

• The integral in this case represents the concept of adding up infinite tiny slices of volume, each shaped like a cylindrical shell.
• Think of it as putting together countless paper-thin layers to create a thick, rich book.
• The integral itself is an expression of calculus that helps us compute these volumes using the method of cylindrical shells.

Thus, integral calculus in this context is all about knitting small bits together into a whole new entity, similar to weaving threads into fabric.

Rotational Solids

Rotational solids emerge beautifully when a plane region rotates around a line, generating a 3D shape.
Imagine swinging a bat in a full circle; the bat's path traces out a solid figure.
Here, our region is beneath the curve of the function \(y = \frac{2}{x}\), between \(x = 1\) and \(x = 2\).

• Think of the cylindrical shells method as a way to find how much space this traced 3D figure occupies.
• When rotated, the function creates shells around the y-axis, with each small vertical slice resembling a hollow cylinder.
• The trick is to see how all these shells stack up, creating the full volumetric form of the solid.

Visualizing this is about grasping how a mere line dance can sculpt something tangible and round.

Definite Integral

A definite integral lets you calculate the net area or "net accumulation" of structures within specified bounds.
In our exercise, \([1, 2]\) forms the bounds.
This gives a finite measure to an infinite process of summation.

• The definite integral computes how the region grows between these critical points.
• Essentially, it captures the entire volume of the rotating region solid within these bounds.
• By evaluating \(\int_{1}^{2} 4\pi \, dx\), you are computing the exact volume of this rotational solid, confirming its finiteness with real limits.

This approach encapsulates the infinite series into a tangible quantity, worth \(4\pi\) in this instance. It is as if you take an endless, fluid motion, pin it down, and measure it precisely.