A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.



a. Use the shell method to write an integral for the volume of the torus.

Verified step by step guidance

1. Identify the region being revolved: The circle of radius 2 centered at (3, 0) lies in the xy-plane and is revolved about the y-axis to form the torus.
2. Set up the shell method: Since the axis of revolution is the y-axis (x = 0), use vertical shells parallel to the y-axis. Each shell corresponds to a vertical slice at a particular x-value between the bounds of the circle.
3. Determine the radius and height of a typical shell: The radius of a shell is the distance from the y-axis, which is simply \(x\). The height of the shell is the vertical length of the circle at that \(x\), which can be found from the circle equation \((x - 3)^2 + y^2 = 2^2\).
4. Express the height of the shell in terms of \(x\): Solve for \(y\) to get \(y = \pm \sqrt{4 - (x - 3)^2}\). The height of the shell is the difference between the top and bottom, so height \(= 2 \sqrt{4 - (x - 3)^2}\).
5. Write the integral for the volume using the shell method formula: \(V = \int_{x=a}^{x=b} 2\pi \times (\text{radius}) \times (\text{height}) \, dx = \int_{1}^{5} 2\pi x \cdot 2 \sqrt{4 - (x - 3)^2} \, dx\), where the limits \(x=1\) and \(x=5\) come from the leftmost and rightmost points of the circle.

Key Concepts

Shell Method for Volume

The shell method calculates volume by integrating cylindrical shells formed by revolving a region around an axis. Each shell's volume is approximated by its circumference, height, and thickness. For revolution about the y-axis, shells are vertical slices parallel to the y-axis, with radius equal to the x-value and height given by the function.

Equation of the Circle and Region Setup

The torus is generated by revolving a circle of radius 2 centered at (3,0) about the y-axis. The circle's equation is (x-3)^2 + y^2 = 4. Understanding this equation helps determine the height of each shell (the vertical distance between the upper and lower parts of the circle) as a function of x.

Limits of Integration and Variable of Integration

When using the shell method about the y-axis, integration is performed with respect to x, ranging over the interval covering the circle's horizontal extent. Here, x varies from 1 to 5 (center 3 minus radius 2 to center 3 plus radius 2). Correct limits ensure the integral accounts for the entire volume of the torus.