Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
86. Find the volume of the solid generated by revolving the region about the x-axis.

Verified step by step guidance

1. Identify the region bounded by the curve \(y = e^{-x}\) and the x-axis in the first quadrant. This means \(x \geq 0\) and \(y \geq 0\).
2. Since the region is revolved about the x-axis, use the disk method to find the volume. The volume of a solid of revolution generated by revolving around the x-axis is given by the integral \(V = \pi \int_a^b [f(x)]^2 \, dx\).
3. Set up the integral limits. Because the region is in the first quadrant and extends infinitely along the x-axis, the limits are from \(0\) to \(\infty\).
4. Write the integral for the volume as \(V = \pi \int_0^{\infty} (e^{-x})^2 \, dx = \pi \int_0^{\infty} e^{-2x} \, dx\).
5. Evaluate the improper integral by finding the antiderivative of \(e^{-2x}\), then apply the limits from \(0\) to \(\infty\) 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 methods like the disk or washer method, which integrate cross-sectional areas perpendicular to the axis of rotation.

Disk Method

The disk method calculates volume by slicing the solid into thin disks perpendicular to the axis of rotation. Each disk's volume is approximated by π(radius)^2 times thickness, and integrating these volumes over the interval gives the total volume.

Exponential Decay Function y = e^(-x)

The function y = e^(-x) represents exponential decay, approaching zero as x increases. Understanding its behavior is essential for setting integration limits and evaluating the integral that defines the volume of the solid formed by revolving the region bounded by this curve and the x-axis.