53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x−x⁴,y=0; about the x-axis.

Verified step by step guidance

1. Identify the region R bounded by the curves: the function \(y = x - x^{4}\) and the line \(y = 0\). This means the region lies between the curve and the x-axis.
2. Determine the interval for \(x\) where the region exists by solving \(x - x^{4} = 0\). Factor the equation as \(x(1 - x^{3}) = 0\), which gives the roots \(x = 0\) and \(x = 1\). So, the region is bounded between \(x = 0\) and \(x = 1\).
3. Since the solid is generated by revolving the region around the x-axis, use the disk method. The volume \(V\) is given by the integral formula: \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\), where \(f(x)\) is the function representing the radius of the disks.
4. In this problem, the radius of each disk is the distance from the x-axis to the curve, which is \(y = x - x^{4}\). Substitute this into the formula to get: \(V = \pi \int_{0}^{1} (x - x^{4})^{2} \, dx\).
5. Expand the integrand \((x - x^{4})^{2}\) and then set up the integral for evaluation. The next step would be to integrate term-by-term over the interval \([0,1]\) 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 can be computed using integral calculus by summing infinitesimal cross-sectional areas perpendicular to the axis of rotation.

Disk and Washer Methods

These methods calculate volume by integrating the area of circular cross-sections. The disk method applies when the region touches the axis of rotation, while the washer method is used when there is a gap, creating hollow sections.

Setting up the Integral with Given Curves

To find the volume, one must express the radius of rotation in terms of x or y using the given functions. For y = x - x⁴ and y = 0, the limits and radius are determined by these curves and the axis of rotation, guiding the integral setup.