Surface area and volume Let f(x) = 1/3 x³ and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2].


b. Find the volume of the solid generated when R is revolved about the y-axis.

Verified step by step guidance

1. Identify the region R bounded by the curve \(f(x) = \frac{1}{3}x^{3}\) and the x-axis on the interval \([0, 2]\). This means the region lies between \(x=0\) and \(x=2\), above the x-axis, and under the curve \(y = \frac{1}{3}x^{3}\).
2. Since the solid is generated by revolving the region R about the y-axis, use the method of cylindrical shells. The formula for the volume using cylindrical shells is: \(V = \int_{a}^{b} 2\pi x \cdot f(x) \, dx\) where \(x\) is the radius of the shell and \(f(x)\) is the height.
3. Substitute the given function \(f(x) = \frac{1}{3}x^{3}\) and the interval limits \(a=0\) and \(b=2\) into the volume integral: \(V = \int_{0}^{2} 2\pi x \cdot \frac{1}{3}x^{3} \, dx\).
4. Simplify the integrand inside the integral: \$2\pi x \cdot \frac{1}{3}x^{3} = \frac{2\pi}{3} x^{4}\(, so the integral becomes \)V = \int_{0}^{2} \frac{2\pi}{3} x^{4} \, dx$.
5. Evaluate the integral by integrating \(x^{4}\) with respect to \(x\), then multiply by the constant \(\frac{2\pi}{3}\), and finally apply the limits from 0 to 2 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. Common methods include the disk/washer method and the shell method. For rotation about the y-axis, the shell method is often more straightforward when the function is given in terms of x.

Shell Method

The shell method calculates volume by summing cylindrical shells formed by revolving vertical slices of the region around the y-axis. Each shell's volume is 2π(radius)(height)(thickness), where radius is the distance from the y-axis, height is the function value, and thickness is a small change in x.

Setting up Definite Integrals

To find the volume, you must express the integral with correct limits and integrand based on the region and axis of rotation. For the shell method, integrate with respect to x over [0, 2], using the radius and height functions derived from the problem's given function and interval.