How to Find Volume by Spinning: Disk Method

This method uses 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.

Recommended EffortlessMath Books

For the calculus background behind volumes of revolution, the Pre-Calculus for Beginners pins down the function and integration prerequisites, and the Calculus for Beginners walks through every integration technique – including disk, washer, and shell methods – with full worked examples.

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Frequently Asked Questions

What is the disk method for finding volumes of revolution?

The disk method finds the volume of a solid formed by spinning a region around an axis. Each thin slice perpendicular to the axis becomes a solid circular disk with volume πr² times its thickness, and an integral adds them all up. It works when the region touches the axis of rotation, so the solid has no hole.

How do you do the disk method?

Sketch the region and the axis, then draw one slice perpendicular to the axis. Its radius is the distance from the axis to the curve, usually f(x). Square the radius, multiply by π and integrate over the bounds. Rotating y = x from 0 to 3 about the x-axis gives V = π∫₀³ x² dx = 9π, the volume of a cone.

What is the disk method formula?

For rotation about the x-axis, V = π∫ from a to b of [f(x)]² dx. For rotation about the y-axis, write the curve as x = g(y) and use V = π∫ from c to d of [g(y)]² dy. For example, rotating y = x² from x = 0 to 1 about the x-axis gives π∫₀¹ x⁴ dx = π/5.

Is the disk method parallel or perpendicular?

Perpendicular. In the disk method, each slice is cut perpendicular to the axis of rotation, so rotating about the x-axis uses vertical slices and dx. The shell method is the one that uses slices parallel to the axis. A quick check: if your thin rectangle touches the axis and spins into a coin, you are using disks.

What is the difference between the disk method and the washer method?

The disk method is for regions that touch the axis, so each slice is a solid disk with one radius. The washer method handles regions with a gap, giving slices with a hole and two radii: V = π∫(R² − r²) dx. When the inner radius r is 0, the washer formula turns into the disk formula, so disks are a special case.

What is the difference between the disk method and the shell method?

Disks slice perpendicular to the axis and use πr² for each slice, while shells slice parallel to the axis and use 2π × radius × height. Both give the same volume for the same solid. Disks are often easier when the radius is a simple function along the axis; shells help when the region is easier to describe in the other variable.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• Volume by spinning: shell method
• Volume by spinning: washer method
• Volume by cross-sections
• How to calculate the volume of cubes and prisms
• How to find the area of a quarter circle

Related: Product Rule of Differentiation · How to Find the Integral of Radicals · Connecting Limits at Infinity and Horizontal Asymptotes