Problem 43 - Thomas Calculus

Question

Find the volume of the torus generated by revolving the circle \((x-2)^{2}+y^{2}=1\) about the \(y\) -axis.

Step-by-Step Solution

Answer

The volume of the torus is calculated using circular cross-sections revolved around the \(y\)-axis.

Step 1: Identify the Elements of the Circle

The given circle equation is \((x-2)^2 + y^2 = 1\). This is a circle centered at \((2, 0)\) with a radius \(r = 1\).

Step 2: Set Up the Volume Integral Using the Washer Method

To find the volume of the torus, we revolve the circle around the \(y\)-axis. The volume \(V\) is found using the washer method:\[V = 2\pi \int_{-1}^{1} (2)^2 - (2 - \sqrt{1 - y^2})^2 \, dy\]

Step 3: Simplify the Expression Inside the Integral

Simplify the expression inside the integral:\[(2)^2 - (2 - \sqrt{1 - y^2})^2 = 4 - [(4 - 4\sqrt{1-y^2} + (1-y^2))] \= 4y^2 - 4\sqrt{1-y^2}\]

Step 4: Evaluate the Integral

Evaluate each part separately if needed or through substitution. Notice this involves trigonometric substitution:\[x = \sin(\theta)\] implies \[dx = \cos(\theta) d\theta\], then process the integral accordingly. Substitution or numerical integration may be needed for detailed steps.

Step 5: Calculate the Volume

Calculate the entire expression to get the volume of the torus. Complete the integration and multiply by any constants. This might require solving a definite integral for verification.

Key Concepts

Washer Method

The washer method is a popular technique used to find the volume of a solid of revolution. It is especially useful when finding the volume of an object with a hole in the middle, like a washer or a torus.

• In essence, you're dealing with a series of concentric disks, which are stacked upon one another to fill the space of the solid.
• The washer method involves subtracting the volume of the inner disk from the outer disk at each point.

The formula for the volume of a solid formed by revolving a region around an axis is given by: \[V = \pi imes ext{(Outer Radius)}^2 - \pi imes ext{(Inner Radius)}^2\]When applied to a torus, like in the given exercise, the outer radius corresponds to the distance from the center of the circle to the axis of revolution plus the radius of the circle. The inner radius is the distance from the center of the circle to the axis of revolution minus the radius of the circle. Using this method, you establish the integral bounds, set up your integral expression, and solve it to find the volume.

Revolution about Y-Axis

When you revolve a shape around an axis, it creates a 3D object. In this case, the exercise involves revolving a circle around the y-axis to form a torus.

• Revolving around the y-axis means every point on the circle describes a circular path, forming a toroidal shape.
• The revolution about the y-axis involves integrating in terms of y, which simplifies the mathematical manipulation when dealing with circles or curves symmetrical along the x-axis.

To set up the integral for revolution about the y-axis, express x in terms of y. For the circle equation \((x-2)^2 + y^2 = 1\), solving it for x provides an expression that helps define inner and outer radii for the washer method. This is crucial in properly setting the bounds and integrands of your volume integral.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots. It leverages trigonometric identities to make the integration process more straightforward.

• This is especially useful when your integrand contains expressions like \(\sqrt{1-y^2}\), as it resembles the Pythagorean identity \(1 = \sin^2(\theta) + \cos^2(\theta)\).
• By setting \(y = \sin(\theta)\), we transform the square root into something manageable.

After making the substitution, the differential \(dy\) converts to \(\cos(\theta)d\theta\). This change of variables simplifies the integral expression considerably, allowing you to perform the integration using familiar trigonometric identities and methods. Finally, remember to replace \(\theta\) with y using the inverse trigonometric function to revert back to the original variable if needed.

Definite Integral

The definite integral is a fundamental concept in calculus used to find the total accumulation of a quantity, such as area or volume, over an interval.

• In our context, it computes the total volume of the torus by integrating along the bounds of the chosen variable, y, from \(-1\) to \(1\).
• Definite integrals consider the limits of integration, unlike indefinite integrals which represent a family of antiderivatives.

When evaluating a definite integral, you'll compute an antiderivative and then apply the Fundamental Theorem of Calculus. This involves plugging in the upper and lower bounds of integration into the antiderivative, and subtracting to find the final result. Ultimately, this allows us to precisely quantify the volume of the torus, offering a robust mathematical approach to solving for the total space occupied by the solid of revolution.