Consider the region bounded by the graphs of
y = arctan(x), y = 0, and x = 1.
b. Find the volume of the solid formed by revolving this region about the y-axis.

Verified step by step guidance

1. First, identify the region bounded by the curves: \(y = \arctan(x)\), \(y = 0\), and the vertical line \(x = 1\). This region lies between \(x=0\) and \(x=1\) since \(y=0\) corresponds to the \(x\)-axis and \(y=\arctan(x)\) is above it for \(x > 0\).
2. Since the solid is formed by revolving the region about the \(y\)-axis, consider using the method of cylindrical shells. The formula for the volume using cylindrical shells when revolving around the \(y\)-axis is: \(V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\)
3. In this problem, the radius of a shell at position \(x\) is the distance from the \(y\)-axis, which is simply \(x\). The height of the shell is the vertical distance between the curves, which is \(\arctan(x) - 0 = \arctan(x)\).
4. Set up the integral for the volume as: \(V = \int_0^1 2\pi x \cdot \arctan(x) \, dx\)
5. To find the volume, you would evaluate the integral \(\int_0^1 x \arctan(x) \, dx\) and then multiply the result by \(2\pi\). Integration by parts is a suitable method to solve this integral.

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. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Choosing the appropriate method depends on the axis of rotation and the given boundaries.

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices around a vertical axis. Each shell's volume is approximated by 2π(radius)(height)(thickness). This method is especially useful when revolving around the y-axis and the function is given in terms of x.

Inverse Trigonometric Functions and Their Properties

Understanding the function y = arctan(x) is crucial, including its domain, range, and behavior. Arctan(x) is the inverse of the tangent function, continuous and increasing for all real x, with horizontal asymptotes at ±π/2. This helps in setting integration limits and interpreting the region bounded by the curves.