Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


b. Repeat part (a) using the disk method.

Verified step by step guidance

1. Identify the region R bounded by the upper half of the circle \(x^{2} + y^{2} = r^{2}\) and the x-axis. Since it is the upper half, the function describing the curve is \(y = \sqrt{r^{2} - x^{2}}\) for \(x\) in \([-r, r]\).
2. Set up the volume integral using the disk method. When revolving around the x-axis, the volume is given by \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\), where \(f(x)\) is the radius of the disk at position \(x\).
3. Substitute \(f(x) = \sqrt{r^{2} - x^{2}}\) into the formula, so the volume integral becomes \(V = \pi \int_{-r}^{r} (\sqrt{r^{2} - x^{2}})^{2} \, dx\).
4. Simplify the integrand: \((\sqrt{r^{2} - x^{2}})^{2} = r^{2} - x^{2}\), so the integral is \(V = \pi \int_{-r}^{r} (r^{2} - x^{2}) \, dx\).
5. Evaluate the definite integral \(\int_{-r}^{r} (r^{2} - x^{2}) \, dx\) by integrating term-by-term and then multiply the result by \(\pi\) to find the volume of the sphere.

Key Concepts

Disk Method

The disk method is a technique for finding the volume of a solid of revolution by slicing the solid perpendicular to the axis of rotation. Each slice forms a disk whose volume is approximated by π(radius)²(thickness). Integrating these volumes over the interval gives the total volume.

Equation of a Circle and Region Definition

The region R is bounded by the upper half of the circle x² + y² = r² and the x-axis, meaning y = √(r² - x²) for x in [-r, r]. Understanding this curve is essential to set up the integral for the volume calculation.

Volume of Revolution about the x-axis

Revolving a region around the x-axis generates a 3D solid. The volume is found by integrating the cross-sectional areas (disks) perpendicular to the x-axis, where each radius corresponds to the y-value of the function defining the region.