Volume of Revolution Calculator

Reviewed by: **David Chen, PhD in Applied Mathematics**.

This calculator employs the Numerical Integration (Trapezoidal Rule) method for solving volumes of revolution, providing a high degree of accuracy for most standard functions.

Welcome to the **Volume of Revolution Calculator**. Use this tool to quickly determine the volume of a solid formed by rotating a function $R(x)$ or the region between $R(x)$ and $r(x)$ around the x-axis, using a numerical approximation method.

Outer Radius Function Squared ($R^2(x)$): Use ‘x’ as the variable. Example: $x^2$ is entered as `x*x`.

Inner Radius Function Squared ($r^2(x)$): Defaults to 0 for the Disk Method. Example: $4-x^2$ revolved about $y=1$ results in $R^2(x) = (4-x)^2$ and $r^2(x)=1^2$.

Lower Bound ($a$):

Upper Bound ($b$):

Number of Sub-Intervals ($N$): Numerical integration steps (e.g., 1000).

Calculated Volume ($V$)

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Detailed Calculation Steps

Steps will appear here after calculation.

Volume of Revolution Calculator Formula (Disk/Washer Method)

$$ V = \pi \int_a^b [R(x)^2 – r(x)^2] dx $$ Where:

$V$ is the Volume of Revolution.
$R(x)$ is the outer radius function.
$r(x)$ is the inner radius function. (For the Disk Method, $r(x) = 0$).
$[a, b]$ is the interval of integration.

Formula Source: Paul’s Online Math Notes | Source 2: Wolfram MathWorld

Variables Explanation

Outer Radius Function Squared ($R^2(x)$): The square of the function that defines the outer edge of the solid being revolved around the axis. Since the integration formula uses the square of the radius, inputting the squared function directly is often simpler.
Inner Radius Function Squared ($r^2(x)$): The square of the function that defines the inner hollow space (for the Washer Method). Use ‘0’ for the Disk Method.
Lower Bound ($a$): The starting x-value of the region being revolved.
Upper Bound ($b$): The ending x-value of the region being revolved.
Number of Sub-Intervals ($N$): The quantity of segments used for the numerical integration. A higher number increases precision.

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What is the Volume of Revolution?

The volume of revolution is a concept in integral calculus used to find the volume of a three-dimensional solid generated by rotating a two-dimensional curve or region around an axis (usually the x or y-axis). These solids, such as cones, spheres, or more complex shapes, are ubiquitous in engineering, physics, and manufacturing.

The primary methods for calculating this volume are the Disk/Washer Method and the Shell Method. The Disk Method applies when the solid is generated by rotating a region adjacent to the axis of revolution, creating thin circular “disks.” The Washer Method is used when the rotated region does not touch the axis, leaving a hole in the center, thus creating a “washer” shape.

How to Calculate Volume of Revolution (Example)

Let’s find the volume of the solid generated by revolving the region under $f(x) = \sqrt{x}$ from $x=0$ to $x=4$ around the x-axis.

Identify the Method: Since the region touches the axis of revolution (x-axis), we use the Disk Method, where $r(x)=0$.
Determine the Radius: The radius $R(x)$ is $\sqrt{x}$. The formula requires $R^2(x)$, so $R^2(x) = (\sqrt{x})^2 = x$.
Define Integration Bounds: The region is bounded by $x=0$ ($a$) and $x=4$ ($b$).
Input Values: Input $R^2(x) = \text{‘x’}$, $r^2(x) = \text{‘0’}$, $a=0$, $b=4$, and $N=1000$ into the calculator.
Apply Formula: The integral becomes $V = \pi \int_0^4 (x – 0) dx$.
Solve (Analytically): The antiderivative of $x$ is $x^2/2$. Evaluating from 0 to 4: $\pi \left[ \frac{4^2}{2} – \frac{0^2}{2} \right] = \pi \left[ 8 – 0 \right] = 8\pi$.
Result: The numerical calculator will approximate this volume to $V \approx 25.1327$.

Frequently Asked Questions (FAQ)

What is the difference between the Disk and Washer Methods?: The Disk Method is a special case of the Washer Method where the region being revolved is adjacent to the axis of revolution, meaning the inner radius, $r(x)$, is zero. The Washer Method is used when there is a gap between the region and the axis, resulting in a hollow solid.
Why does this calculator use numerical integration?: Calculators that handle arbitrary functions on the web must rely on numerical methods (like the Trapezoidal Rule) because symbolic integration—finding the exact antiderivative of any function—is computationally complex and unreliable for client-side JavaScript execution.
What are the inputs $R^2(x)$ and $r^2(x)$?: These are the squares of the radii functions, $R(x)$ and $r(x)$. The volume formula naturally squares the radius before integration, $V = \pi \int (R^2 – r^2) dx$. By asking for the squared value, we simplify the user input and reduce potential JavaScript complexity.
How do I calculate the volume of revolution around the y-axis?: To revolve around the y-axis, you must express your functions in terms of $y$ ($x=g(y)$ and $x=h(y)$), and the integration bounds must be $y$-values ($c$ to $d$). The formula becomes $V = \pi \int_c^d [R(y)^2 – r(y)^2] dy$.