How to Find Volume by Spinning: Disk Method

This method leverages the symmetry of revolving shapes, transforming a 2D area into a 3D object. Each disk’s volume is \( \pi r^2 \Delta x \), where \( r \) is the radius function. By accumulating these volumes along the axis, complex volumetric problems are simplified. Especially useful in engineering and design, the Disk Method provides a foundational tool for calculating storage capacities, structural volumes, and more.

Here are the steps to use the method:

1. Define the Radius: The radius \(r\) of each disk is a function of \(x\) (or \(y\), depending on the axis of revolution), denoted as \(r(x)\) or \(r(y)\).
2. Disk Volume: The volume of each infinitesimally thin disk is given by the formula \(dV = \pi [r(x)]^2 dx\) (if revolving around the y-axis) or \(dV = \pi [r(y)]^2 dy\) (if revolving around the x-axis), where \(dV\) is the differential volume of the disk.
3. Set Up the Integral: The total volume \(V\) of the solid is the integral of these differential volumes over the interval of revolution. Formally, if revolving around the y-axis:
   \( V = \int_{a}^{b} \pi [r(x)]^2 dx \)
   And if revolving around the x-axis:
   \( V = \int_{c}^{d} \pi [r(y)]^2 dy \)
   Here, \(a\) and \(b\) (or \(c\) and \(d\)) are the bounds of the region being revolved, determined by the intersection points of the curve and the axis of revolution or the specified limits of integration.
4. Evaluate the Integral: Compute the integral to find the total volume of the solid of revolution. This step may require analytical techniques or numerical methods depending on the function defining the radius.

By applying this method, you can calculate the volume of a wide variety of shapes with rotational symmetry, providing a powerful technique for analyzing physical objects and theoretical constructs in mathematics, physics, and engineering.

Let’s calculate the volume of a solid formed by revolving the region under the curve \( y = \sin(x) \) from \( x = 0 \) to \( x = \pi \) around the x-axis.

Step-by-Step Solution:

1. Define the Radius:

• The radius of each disk is given by the function \( y = \sin(x) \).

2. Disk Volume:

• The volume of each disk is \( dV = \pi [r(x)]^2 dx = \pi [\sin(x)]^2 dx \).

3. Set Up the Integral:

• The total volume \( V \) is the integral of \( \pi [\sin(x)]^2 \) from \( x = 0 \) to \( x = \pi \):
  \( V = \int_{0}^{\pi} \pi [\sin(x)]^2 dx \)

4. Simplify the Integral Using a Trigonometric Identity:

• Use the identity \( \sin^2(x) = \frac{1 – \cos(2x)}{2} \) to simplify the integral:
  \( V = \pi \int_{0}^{\pi} \frac{1 – \cos(2x)}{2} dx \)

The volume of the solid formed by revolving the region under the curve \( y = \sin(x) \) from \( x = 0 \) to \( x = \pi \) around the x-axis is approximately \(4.93\) cubic units.