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 = x³,y=0, and x=2; about the x-axis

Verified step by step guidance

1. Identify the region R bounded by the curves: \(y = x^{3}\), \(y = 0\), and \(x = 2\). This region lies between the curve \(y = x^{3}\) and the x-axis from \(x = 0\) to \(x = 2\).
2. Since the region is revolved about the x-axis, use the disk or washer method. Here, the cross-sectional disks are perpendicular to the x-axis, and the radius of each disk is given by the function value \(y = x^{3}\).
3. Write the formula for the volume of the solid using the disk method: \(V = \pi \int_{a}^{b} [R(x)]^{2} \, dx\), where \(R(x)\) is the radius of the disk at position \(x\). In this case, \(R(x) = x^{3}\), and the limits of integration are from \(a = 0\) to \(b = 2\).
4. Set up the integral for the volume: \(V = \pi \int_{0}^{2} (x^{3})^{2} \, dx = \pi \int_{0}^{2} x^{6} \, dx\).
5. Evaluate the integral by finding the antiderivative of \(x^{6}\), then apply the limits of integration from 0 to 2, and multiply the result by \(\pi\) 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 are techniques to calculate volumes of solids of revolution. The disk method uses circular cross-sections when the region touches the axis of rotation, while the washer method applies when there is a hole, subtracting inner radius areas from outer radius areas.

Setting up Definite Integrals with Given Bounds

To find the volume, one must correctly identify the limits of integration from the region's boundaries and express the radius of rotation as a function of the variable of integration. Here, the bounds are from x=0 to x=2, and the radius is determined by y = x³.