How to find volume using cylindrical shells

A strip parallel to the axis of rotation sweeps out a thin cylindrical shell. Its radius tells us how far it travels around the axis, and its height comes from the length of the strip. We’ll use those two measurements to build the volume integral.

Learn Calculus 2 with meStep-by-step videos, lesson notes, and practice for integration, applications, and more.

Explore Calculus 2 →

Multiply circumference, height, and thickness

One shell uses its circumference, its height, and its small thickness.

The volume of a thin shell is approximately its circumference times its height times its thickness. Add up the shells with an integral. The radius is a distance to the axis, and the height is the distance between the curves along the strip.

Use vertical shells around the y-axis

For the region between $y=x^3$ and the $x$-axis on $[1,2]$, the radius is $x$ and the height is $x^3$.

$\begin{array}{l}V=2\pi {\int }_1^{2}x\left(x^3\right)dx\\ =\frac{2\pi }{5}(2^5-1^5)=\frac{62\pi }{5}\end{array}$

Use horizontal shells around the x-axis

For the region between $x=3y^4$ and the $y$-axis on $2\le y\le 4$, the radius is $y$ and the height is $3y^4$.

$\begin{array}{l}V=2\pi {\int }_2^{4}y\left(3y^4\right)dy\\ =\pi (4^6-2^6)=4032\pi \end{array}$

Full Calculus 2 course

Learn Calculus 2 with me

We’ll work through integration, its applications, and series, with the steps explained as we go.

Explore my Calculus 2 course →

• Video lessons
• Lesson notes
• Practice problems

Included with membership, along with every other Krista King Math course.