June 8, 2026 - Calculus 2 - Cylindrical Shell Method Lecture Notes

Flashcards covering the definitions, formulas, and examples of the cylindrical shell method used to calculate volumes of solids of revolution.

Cylindrical Shell Method

A technique for finding the volume of a solid of revolution by integrating the surface areas of nested hollow cylinders, typically used to keep the integration in terms of $x$ when revolving around the $y$-axis.

Shell Circumference

The length of a shell when sliced and unfolded, calculated as $2\times\text{pi}\times\text{r}$, where the radius for a rotation around the $y$-axis is the horizontal distance $x$.

Shell Height

The vertical dimension of the cylindrical shell, determined by the function $f(x)$ or the difference between two functions $f(x)-g(x)$.

Shell Thickness

The width of the rectangle used to generate the shell, represented by the differential element $dx$.

Shell Method Volume Formula ($y$-axis)

The integral used to find the volume of a solid revolved around the $y$-axis, expressed as $\int_{a}^{b} 2\times\text{pi}\times x\times f(x)\,dx$.

Volume of $y=\frac{1}{x}$ from $1$ to $3$

The solid of revolution obtained by spinning $y=\frac{1}{x}$ around the $y$-axis, resulting in a volume of $4\times\text{pi}$ units cubed.

Vertex of a Parabola

The point located at the peak of the graph, found by calculating the opposite of $B$ over $2\times A$ ($-\frac{b}{2a}$) to find the $x$-coordinate.

Shell Method Volume Formula ($x$-axis)

The integral used when revolving around the $x$-axis where the radius is $y$ and the height is a function of $y$, expressed as $\int_{c}^{d} 2\times\text{pi}\times y\times g(y)\,dy$.

Negative Volume Error

An issue that occurs when a function crosses the $x$-axis (like $\cos(x)$ on $[0, \text{pi}]$), requiring the integral to be broken into two separate intervals to ensure positive area.

Cross-section Method

An alternative to the shell method where volume is found by integrating the area of slices, such as squares, perpendicular to the $x$-axis.