43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated? 

Verified step by step guidance

1. First, identify the region bounded by the curves \(y = \sec x\) and \(y = 2\) on the interval \$0 \leq x \leq \frac{\pi}{3}\(. Since \)\sec x = \frac{1}{\cos x}\(, note that \)\sec x\( is increasing on this interval and \)\sec 0 = 1\(, \)\sec \frac{\pi}{3} = 2\(, so the curves intersect at \)x = \frac{\pi}{3}$.
2. Since the region is revolved about the x-axis, consider the vertical slices perpendicular to the x-axis. The volume can be found using the washer method because the region lies between two curves above the x-axis.
3. Set up the volume integral using the washer method formula: \(V = \pi \int_a^b \left(R(x)^2 - r(x)^2\right) \, dx\), where \(R(x)\) is the outer radius and \(r(x)\) is the inner radius of the washers.
4. Determine the outer and inner radii. The outer radius \(R(x)\) is the distance from the x-axis to the upper curve \(y=2\), so \(R(x) = 2\). The inner radius \(r(x)\) is the distance from the x-axis to the lower curve \(y=\sec x\), so \(r(x) = \sec x\).
5. Write the integral explicitly: \(V = \pi \int_0^{\frac{\pi}{3}} \left(2^2 - (\sec x)^2\right) \, dx = \pi \int_0^{\frac{\pi}{3}} \left(4 - \sec^2 x\right) \, dx\). This integral can then be evaluated to find the volume.

Key Concepts

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The volume is computed by integrating cross-sectional areas perpendicular to the axis of rotation, using methods like disks, washers, or shells.

Disk and Washer Methods

The disk method calculates volume by summing up circular cross-sections (disks) when the region touches the axis of rotation. The washer method extends this by subtracting inner holes, useful when the region is bounded by two curves, creating hollow cross-sections.

Choosing the Axis and Limits of Integration

Identifying the axis of rotation (here, the x-axis) and the interval for integration (0 to π/3) is crucial. This determines the shape of cross-sections and the bounds for the integral that computes the volume of the solid.