Volume of Solid of Revolution

Volumes of Revolution
Integral Calculus » Centroids | Volumes and Surface Areas » Integration by Parts

What is a Solid of Revolution?

A solid of revolution is a three-dimensional figure obtained by rotating a two-dimensional plane region around a straight line (the axis of revolution) that lies in the same plane.

Imagine taking a flat shape, like the area under a curve, and spinning it around an axis. The path swept out by this area forms a solid object. For example:

• Rotating a rectangle around one of its sides generates a cylinder.
• Rotating a right triangle around one of its legs generates a cone.
• Rotating a semicircle around its diameter generates a sphere.

Calculus, specifically integration, provides powerful methods to calculate the volume of these solids, even when the generating region is bounded by complex curves.

Disk Method (Rotation about the x-axis)

The Disk Method is used to find the volume of a solid of revolution when the region being revolved is bounded by a function $y=f(x)$, the x-axis (i.e., $y=0$), and vertical lines $x=a$ and $x=b$, and this region is rotated around the x-axis.

We can think of slicing the solid into infinitesimally thin circular disks perpendicular to the axis of rotation (the x-axis in this case).

• Each disk has a radius $R(x) = f(x)$ (assuming $f(x) \ge 0$).
• The area of the face of such a disk is $A(x) = \pi [R(x)]^2 = \pi [f(x)]^2$.
• The thickness of each disk is $dx$.
• The volume of an infinitesimal disk is $dV = A(x) \, dx = \pi [f(x)]^2 \, dx$.

To find the total volume $V$, we sum (integrate) the volumes of these disks from $x=a$ to $x=b$:

$$ V = \int_a^b \pi [f(x)]^2 \, dx = \pi \int_a^b [f(x)]^2 \, dx $$

This formula applies when the region is flush against the axis of rotation.

Washer Method (Rotation about the x-axis)

The Washer Method is an extension of the Disk Method. It is used when the region being revolved is bounded by two functions, an outer function $y=R(x)$ and an inner function $y=r(x)$ (where $R(x) \ge r(x) \ge 0$ on $[a,b]$), and this region is rotated around the x-axis.

When this region is revolved, it forms a solid with a hole in the middle, resembling a washer.

• The outer radius of the washer is $R(x)$.
• The inner radius of the washer (the radius of the hole) is $r(x)$.
• The area of the face of such a washer is $A(x) = \pi ([R(x)]^2 - [r(x)]^2)$.
• The thickness of each washer is $dx$.
• The volume of an infinitesimal washer is $dV = A(x) \, dx = \pi ([R(x)]^2 - [r(x)]^2) \, dx$.

To find the total volume $V$, we integrate from $x=a$ to $x=b$:

$$ V = \int_a^b \pi ([R(x)]^2 - [r(x)]^2) \, dx = \pi \int_a^b \left( [R(x)]^2 - [r(x)]^2 \right) \, dx $$

Practical Applications

Calculating volumes of solids of revolution has numerous applications in various fields:

• Engineering and Manufacturing: Designing tanks, pipes, calculating material for machine parts.
• Architecture: Designing domes, columns, estimating material volumes.
• Physics: Calculating moments of inertia, fluid dynamics problems.
• Medicine: Modeling volumes of organs.
• Aerospace: Designing rocket nozzles and fuel tanks.