Shell Method Calculator

What is a Shell Method Calculator?

A Shell Method Calculator estimates or exactly evaluates volumes of solids of revolution by summing the volumes of thin cylindrical shells. When a region in the plane is revolved about a vertical or horizontal axis, each shell contributes circumference \((2\pi r)\) times height \((h)\) times thickness \((\Delta x\ \text{or}\ \Delta y)\). Unlike the washer/disc method (which stacks cross-sections perpendicular to the axis), the shell method integrates parallel to the axis, making it especially convenient when functions are easier to express as “top minus bottom” or “right minus left” along shells. The calculator builds the correct radius and height, chooses \(dx\) or \(dy\) according to the axis, sets bounds from intersections, and returns a clean integral with steps.

About the Shell Method Calculator

For rotation about a vertical line \(x=a\), shells are vertical (use \(x\)-integration). The radius is the horizontal distance to the axis, \(r(x)=|x-a|\), and the shell height is the vertical span \(h(x)=y_{\text{top}}(x)-y_{\text{bot}}(x)\). For rotation about a horizontal line \(y=b\), shells are horizontal (use \(y\)-integration). The radius is \(r(y)=|y-b|\), and the height is the horizontal span \(h(y)=x_{\text{right}}(y)-x_{\text{left}}(y)\). The tool symbolically simplifies the integrand, handles piecewise regions, and warns about sign issues (e.g., swapped top/bottom). It also supports offset axes (e.g., around \(x=3\) or \(y=-2\)), and can provide numeric values if the antiderivative is non-elementary.

How to Use this Shell Method Calculator

1. Specify the region: curves (top/bottom or left/right) and intersection bounds.
2. Choose the axis of rotation (e.g., \(y\)-axis, \(x\)-axis, \(x=a\), \(y=b\)).
3. The calculator selects \(dx\) or \(dy\) to make shells parallel to the axis and constructs \(r\) and \(h\).
4. Review the shell integral, compute the antiderivative, and evaluate at bounds for the final volume.
5. Compare with the washer method if desired; both should match when set up correctly.

Core Formulas (LaTeX)

Vertical shells (about a vertical line \(x=a\)): \[ V = 2\pi \int_{x_1}^{x_2} r(x)\,h(x)\,dx \quad\text{with}\quad r(x)=|x-a|,\ \ h(x)=y_{\text{top}}(x)-y_{\text{bot}}(x). \]

Horizontal shells (about a horizontal line \(y=b\)): \[ V = 2\pi \int_{y_1}^{y_2} r(y)\,h(y)\,dy \quad\text{with}\quad r(y)=|y-b|,\ \ h(y)=x_{\text{right}}(y)-x_{\text{left}}(y). \]

Examples (Illustrative)

Example 1 — \(y=x\) on \([0,1]\) about the \(y\)-axis

\(r(x)=x,\ h(x)=x\). \(\displaystyle V=2\pi\int_0^1 x\cdot x\,dx =2\pi\int_0^1 x^2\,dx =2\pi\left[\frac{x^3}{3}\right]_0^1 =\frac{2\pi}{3}.\)

Example 2 — Region between \(y=x\) and \(y=x^2\) on \([0,1]\), about the \(y\)-axis

\(h(x)=x-x^2,\ r(x)=x\). \(\displaystyle V=2\pi\int_0^1 x(x-x^2)\,dx =2\pi\int_0^1 (x^2-x^3)\,dx =2\pi\left(\frac{1}{3}-\frac{1}{4}\right) =\frac{\pi}{6}.\)

Example 3 — Under \(y=\sqrt{x}\), \(0\le x\le 4\), about the \(x\)-axis

Use horizontal shells: \(x=y^2\), \(0\le y\le 2\). Then \(r(y)=y\), \(h(y)=y^2-0\). \(\displaystyle V=2\pi\int_0^2 y(y^2)\,dy =2\pi\int_0^2 y^3\,dy =2\pi\left[\frac{y^4}{4}\right]_0^2 =8\pi.\)

Example 4 — Triangle \(y=2x\), \(y=0\), \(0\le x\le 1\), about \(x=3\)

Vertical shells: \(h(x)=2x\), \(r(x)=|x-3|=3-x\). \(\displaystyle V=2\pi\int_0^1 (3-x)(2x)\,dx =2\pi\int_0^1 (6x-2x^2)\,dx =2\pi\left(3-\frac{2}{3}\right) =\frac{14\pi}{3}.\)

FAQs

When should I use shells instead of washers/discs?

Use shells when “top–bottom” or “right–left” is simple and the axis is parallel to those spans, giving an easier integral.

How do I pick \(dx\) or \(dy\)?

Choose the variable so shells run parallel to the axis: vertical shells \(\Rightarrow dx\); horizontal shells \(\Rightarrow dy\).

What if my axis is offset, like \(x=a\) or \(y=b\)?

Replace the radius by the distance to that line: \(r(x)=|x-a|\) or \(r(y)=|y-b|\).

How do I find bounds?

Intersect the boundary curves. For \(dx\), use \(x\)-values; for \(dy\), use \(y\)-values.

What if “top” and “bottom” swap in the interval?

Split the integral at intersection points so each subinterval uses a consistent \(h=\text{top}-\text{bottom}\).

Do shells always match washers?

Yes, when set up correctly both methods produce the same volume; they are different geometric decompositions of the same solid.