Question

Set up the integral (using shells) for the volume of the torus obtained by revolving the region inside the circle \(x^{2}+y^{2}=a^{2}\) about the line \(x=b\), where \(b>a\). Then evaluate this integral. Hint: As you simplify, it may help to think of part of this integral as an area.

Short answer

The volume of the torus is \(4\pi^2 ba^2\).

Step-by-step solution

Understand the Problem

We need to find the volume of a torus generated by revolving a circle with equation \(x^2 + y^2 = a^2\) about the line \(x = b\), where \(b > a\). The method of cylindrical shells will be used for this purpose.

Set Up Limits and Radius for the Cylindrical Shells

A typical shell is obtained by considering a vertical strip at position \(x\) from the circle center to its edge. The limits for \(x\) are from \(-a\) to \(a\) because the circle has radius \(a\). The radius of the shell relative to the axis of revolution is \(b - x\) as the distance between the line \(x = b\) and the strip at \(x\).

Find the Height of the Cylindrical Shells

The height of each shell is determined by the portion of the circle at position \(x\). Using the circle equation \(x^2 + y^2 = a^2\), we find the extent in the y-direction, which is \(2 \sqrt{a^2 - x^2}\).

Set Up the Integral for Volume Using Shell Method

The volume \(V\) of the shell is given by the integral formula \(V = \int_{-a}^{a} 2\pi (\text{radius}) \times \text{height} \, dx\). Substituting for radius and height, we get: \[ V = \int_{-a}^{a} 2\pi (b-x) \cdot 2 \sqrt{a^2 - x^2} \, dx \].

Simplify and Evaluate the Integral

The volume integral becomes:\[ V = 4\pi \int_{-a}^{a} (b-x) \sqrt{a^2 - x^2} \, dx \].Splitting into two integrals:\[ V = 4\pi \left(b \int_{-a}^{a} \sqrt{a^2 - x^2} \, dx - \int_{-a}^{a} x \sqrt{a^2 - x^2} \, dx \right). \]The first integral represents half a circle's area: \( \int_{-a}^{a} y \, dx = \pi a^2 \). The second integral evaluates to zero due to symmetry:\[ V = 4\pi \left(b \pi a^2 - 0\right) = 4\pi^2 ba^2 \].

Conclusion

The volume of the torus is \(4\pi^2 ba^2\), obtained by evaluating the integral using the method of cylindrical shells.

Key concepts

Torus Volume

Understanding the volume of a torus involves visualizing a donut-shaped solid that arises from rotating a circle around an axis outside the circle, such as the line \(x = b\). This structure has a hole in its center, much like a ring or a tire. To calculate the volume, we think of building the torus by stacking numerous thin cylindrical shells around the center axis. These shells are generated as the circle rotates around the line \(x=b\). The key to finding the torus volume is using the shell method to add up the volume of each of these cylinders. We compute the total volume by integrating over the entire region the circle occupies. This technique is crucial for understanding how rotation of a simple 2D shape can create complex 3D geometries.

Circle Equation

A circle is defined in the Cartesian plane by the equation \(x^2 + y^2 = a^2\). Here, \(a\) represents the circle's radius, which is the distance from the center of the circle to any point on its edge.This simple equation tells us a lot: every point \((x, y)\) on the circle is exactly \(a\) units from the center. This distance being consistent in all directions ensures the perfect roundness of the circle. When used in rotation, this form helps us define every point that will sweep out to form a torus. Knowing this equation is fundamental in applications involving rotary volumes, such as when calculating torus shapes or other related geometry problems.

Shell Method

The shell method is a powerful tool in integral calculus for computing volumes of revolution. Imagine slicing our shape into thin, circular tubes or shells rather than discs, which may be more typical.The idea is straightforward: each cylindrical shell has a radius, a height, and a thickness (expressed as \(dx\), an infinitesimally small change in \(x\)). We find the volume of each shell by multiplying these aspects:

• Radius: The distance from the axis of rotation (line \(x=b\)) to the position \(x\) of the shell, hence \(b-x\).
• Height: Determined by the maximum extent of the circle along the \(y\)-axis, calculated as \(2 \sqrt{a^2 - x^2}\).
• Thickness: Represented by \(dx\) to account for all minuscule segments.

Integrating these over the interval from \(-a\) to \(a\) gives us the total volume. This method is particularly effective for shapes like tori, where the axis of rotation is external to the figure.

Integral Calculus

Integral calculus is all about accumulation and summing up countless tiny bits to comprehend the whole. It's employed in various applications, including physics, economics, and in this context, geometry, to solve real-world problems.The integral \(\int_{-a}^{a} (b-x) \sqrt{a^2 - x^2} \, dx\) represents the essence of adding up volumes of the shells to find the torus's total volume. Through careful analysis, integrals handle areas, volumes, and numerous other quantities that comprise small infinitesimals.Two key concepts in integral calculus apply here:

• Definite integrals: Calculate the total from a starting point to an end point. Here, they help sum up the cylindrical shells from \(-a\) to \(a\).
• Substitution and simplification: Simplifying elements of the integral, often using substitution or recognizing symmetries, makes integration manageable, as seen in setting the integral for the torus volume.

Integral calculus allows us to explore the unseen depths of mathematical curves and revolve them to comprehend their 3D potential.