Problem 44 - Thomas Calculus

Question

Find the volume of the solid generated by revolving each region about the \(y\) -axis. \begin{equation} \begin{array}{l}{\text { The region in the first quadrant bounded on the left by the circle }} \\ {x^{2}+y^{2}=3, \text { on the right by the line } x=\sqrt{3}, \text { and above by the }} \\ {\text { line } y=\sqrt{3}}\end{array} \end{equation}

Step-by-Step Solution

Answer

The volume of the solid is \(\pi (3\sqrt{3} - 2)\).

Step 1: Understand the Region

The region is bounded by the circle \(x^2 + y^2 = 3\), the line \(x = \sqrt{3}\), and \(y = \sqrt{3}\). All boundaries are in the first quadrant. The circle is centered at the origin with radius \(\sqrt{3}\). The vertical line \(x = \sqrt{3}\) and the horizontal line \(y = \sqrt{3}\) complete the region.

Step 2: Set Up the Integral for Volume

The solid is formed by revolving the specified region around the \(y\)-axis. This requires the method of cylindrical shells. The formula used is \( V = \int_{a}^{b} 2\pi x f(x) \, dx \), where \(f(x)\) is the height of the shell. The bounds of integration are from where the circle intercepts the x-axis, \(x = 0\), to \(x = \sqrt{3}\).

Step 3: Determine Height of Each Shell

The height of each cylindrical shell is given by the formula \(f(x) = \sqrt{3} - \sqrt{3 - x^2}\). The top of the region is line \(y = \sqrt{3}\) and the bottom is the intersection of the circle \(x^2 + y^2 = 3\), solved for \(y\), as \(y = \sqrt{3 - x^2}\).

Step 4: Compute the Volume Integral

Substitute \(f(x)\) into the volume formula resulting in \[ V = \int_{0}^{\sqrt{3}} 2\pi x (\sqrt{3} - \sqrt{3 - x^2}) \, dx \]This integral can be split as\[ V = 2\pi \left( \int_{0}^{\sqrt{3}} x\sqrt{3} \, dx - \int_{0}^{\sqrt{3}} x\sqrt{3 - x^2} \, dx \right) \]

Step 5: Integrate Each Term Separately

The first term is straightforward. Compute \[ \int_{0}^{\sqrt{3}} x\sqrt{3} \, dx = \sqrt{3} \left[ \frac{x^2}{2} \right]_{0}^{\sqrt{3}} = \frac{3\sqrt{3}}{2} \] For the second term, the substitution \(u = 3 - x^2\) (\(du = -2x \, dx\)) simplifies the integral:\[ \int_{0}^{\sqrt{3}} x\sqrt{3-x^2} \, dx = -\frac{1}{2} \int_{3}^{0} \sqrt{u} \, du = \frac{1}{2} \int_{0}^{3} \sqrt{u} \, du \]

Step 6: Evaluate the Substitution Integral

The integral \(\frac{1}{2} \int_{0}^{3} \sqrt{u} \, du\) evaluates as follows:\[ \frac{1}{2} \left[ \frac{2}{3}u^{3/2} \right]_{0}^{3} = \frac{1}{3} [u^{3/2}]_{0}^{3} = \frac{1}{3}(3) = 1 \]

Step 7: Calculate Final Volume

Substituting back, we have:\[ V = 2\pi \left( \frac{3\sqrt{3}}{2} - 1 \right) = 2\pi \left( \frac{3\sqrt{3} - 2}{2} \right) = \pi (3\sqrt{3} - 2) \]

Key Concepts

Cylindrical Shells

To find the volume of a solid formed by rotating a region around an axis, we can use the cylindrical shells method. This method is especially useful when the solid is generated by revolving a region around a vertical line, such as the y-axis. A cylindrical shell is essentially a hollow tube that forms as the region moves around the axis.

The idea is to divide the region into thin vertical strips. Each strip, when revolved around the axis, creates a cylindrical shell. The radius of each shell corresponds to the x-coordinate of the strip, while the height of the shell is determined by the function that describes the upper boundary of the region.

The formula for the volume of a cylindrical shell is:

• Volume = \( 2\pi x f(x) \triangle x \)

where \( x \) is the distance from the y-axis (radius), \( f(x) \) is the height of the shell, and \( \triangle x \) is the thickness of the shell or strip. When the thickness approaches zero, the integral over all shells provides the total volume.

Definite Integral

The concept of a definite integral plays a crucial role in calculating the volume of a solid of revolution. When we integrate a function over an interval, we are essentially summing up an infinite number of infinitely small quantities, which, in this context, are the volumes of the thin shells.

In the cylindrical shells method, the definite integral is used to add up the volumes of all the cylindrical shells from one boundary of the region to the other. The limits of the integral correspond to the points where the region starts and ends along the x-axis. For this problem, the bounds are from 0 to \( \sqrt{3} \).

The integral in its complete form is expressed as:

• \( V = \int_{a}^{b} 2\pi x f(x) \, dx \)

Evaluating this definite integral yields the total volume of the solid.

Volume Integral

The term "volume integral" refers to the integral expression used to find the overall volume of a solid. In the method of cylindrical shells, the volume integral is constructed based on the geometric properties of each shell.

The integral combines the radius and height formulas to integrate over the required interval. In the exercise, the region to be revolved encompasses the bounded area by a circle and lines. Determining the height correctly involves understanding these geometric constraints.

The volume integral is typically evaluated by separating into simpler components like:

• \( V = 2\pi \left( \int_{a}^{b} x f_1(x) \, dx - \int_{a}^{b} x f_2(x) \, dx \right) \)

This approach allows for easier computation by handling each relevant part of the function independently, thus simplifying the final integral evaluation.

Revolving Around Y-Axis

When a region is revolved around the y-axis, it results in the formation of a three-dimensional shape with symmetrical properties. This axis-specific revolution makes using the cylindrical shells method particularly suitable.

Revolving around the y-axis implies that each point in the region traces a circular path at a certain distance from the axis. This distance plays directly into the formula as the radius of the shell. As the axis is vertical, we observe horizontal strips forming the heights of the shells.

The significance of the y-axis revolution includes:

• The radius of each cylindrical shell is determined by the x-coordinate.
• The height comes from the vertical distance between intersecting curves.
• The total volume is obtained by integrating these shells from left to right boundaries of the region, defined along the x-axis.

Ultimately, revolving around the y-axis results in a methodical setup for evaluating the volume through cylindrical integration.