Problem 33 - Thomas Calculus

Question

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(y\)-axis. \(x=2 / \sqrt{y+1}, \quad x=0, \quad y=0, \quad y=3\)

Step-by-Step Solution

Answer

The volume is \( 8\pi \ln(2) \).

Step 1: Understand the Problem

We need to find the volume of the solid generated by revolving the region bounded by the given curves and lines around the y-axis. The boundaries are defined by the curves and lines: \( x = \frac{2}{\sqrt{y+1}} \), \( x = 0 \), \( y = 0 \), and \( y = 3 \).

Step 2: Set Up the Integral for Volume

We use the method of disks or washers to set up the integral. The volume \( V \) of the solid can be calculated using the formula: \[ V = \pi \int_{a}^{b} (f(y))^2 \, dy \] where \( f(y) = 2 / \sqrt{y+1} \) for the given problem, \( a = 0 \), and \( b = 3 \).

Step 3: Evaluate the Integral

Substitute \( f(y) = 2 / \sqrt{y+1} \) into the volume integral and simplify:\[V = \pi \int_{0}^{3} \left( \frac{2}{\sqrt{y+1}} \right)^2 \, dy = \pi \int_{0}^{3} \frac{4}{y+1} \, dy.\]Simplify and evaluate the integral:\[V = 4\pi \left[ \ln|y+1| \right]_{0}^{3} = 4\pi \left( \ln(4) - \ln(1) \right) = 4\pi \ln(4).\]

Step 4: Simplify the Result

Now simplify the expression further if needed: \[V = 4\pi \ln(4) = 8\pi \ln(2).\] This is the expression for the volume of the solid revolved around the y-axis.

Key Concepts

Disk Method

The Disk Method is a technique used in calculus to find the volume of a solid of revolution. It involves revolving a region around a line (often the x-axis or y-axis) to create a solid shape.In this method, the solid is thought of as being made up of a series of thin disks stacked together.To find the volume of one of these disks, we use the formula \[V = \pi (\text{radius})^2 \times \text{thickness},\]where the radius is the distance from the axis of revolution to the boundary of the region and the thickness is an infinitesimally small difference, usually represented as \(dy\) or \(dx\).

• If revolving around the x-axis, use \(dx\)
• If revolving around the y-axis, use \(dy\)

In our example, we use the disk method revolving around the y-axis. The function \(x = \frac{2}{\sqrt{y+1}}\) gives us the radius, and we integrate over the y-range from 0 to 3.This method is particularly useful when the radius of each disk is expressed as a function of either x or y.

Integral Calculus

Integral calculus is a branch of calculus that deals with the concept of integrals.It provides tools for finding volumes, areas, and other quantities that accumulate across a certain domain.In our problem, we use integral calculus to set up and evaluate integrals that represent the volume of a solid of revolution.The integral \[ \pi \int_{a}^{b} (f(y))^2 \, dy\]is central to our calculation.This integral sums up the volume of infinitesimally thin disks from \(y = a\) to \(y = b\), effectively building up the volume of the entire solid.

• The function \(f(y)\) represents the radius of each disk.
• The boundaries \(a\) and \(b\) are determined by the constraints of the problem—in this case, from 0 to 3.

Evaluating the integral requires applying the fundamental theorem of calculus, which connects differentiation with integration, allowing us to compute definite integrals.

Solid of Revolution

A solid of revolution is a three-dimensional shape formed by rotating a two-dimensional region around a line (the axis of revolution).In this exercise, the rotation takes place around the y-axis, transforming a flat area bounded by curves into a solid form.The original region in our example is defined by the curve \(x = \frac{2}{\sqrt{y+1}}\), and lines at \(x = 0\), \(y = 0\), and \(y = 3\).By revolving this region around the y-axis, a solid is created, which resembles a three-dimensional donut or a vase.To find the volume of such a solid, methods like the disk or washer method are employed to address the symmetric nature of revolution.

• Revolutions around different axes or different functions can result in varied solid shapes.
• Understanding the initial region and its limits is crucial for applying calculus methods effectively.

Thus, the solid of revolution not only encompasses complex geometries but also showcases the beauty of calculus in calculating practical geometric attributes.