Use this powerful tool to calculate the volume of the solid generated when a region under a curve is revolved around the x-axis, applying the Disk Method with numerical integration for accuracy.
Volume of Solid of Revolution Calculator
Calculated Volume ($V$)
Volume of Solid of Revolution Formula (Disk Method)
$$V = \pi \int_{a}^{b} [f(x)]^2 dx$$
Formula Source: Wolfram MathWorld, Paul’s Online Notes
Variables Explained
- **Function $f(x)$:** The equation representing the curve that forms the boundary of the region. This is the radius of the disk at any point $x$.
- **Lower Limit, $a$:** The starting x-coordinate of the region being revolved.
- **Upper Limit, $b$:** The ending x-coordinate of the region being revolved.
- **$V$:** The resulting Volume of the Solid of Revolution.
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What is the Volume of a Solid of Revolution?
A Solid of Revolution is a three-dimensional shape formed when a two-dimensional area, or region, is rotated around a line (the axis of revolution). This concept is fundamental in integral calculus and is used to determine the volume of many complex, rotationally symmetric shapes that are difficult to measure using standard geometric formulas.
The two primary methods for calculating this volume are the Disk/Washer Method and the Shell Method. The Disk Method, used in this calculator, involves slicing the resulting solid perpendicular to the axis of revolution (in this case, the x-axis) into infinitesimally thin disks. The volume of each disk is calculated as the area of the circular face ($\pi r^2$) multiplied by the thickness ($dx$).
Summing these infinitesimal disk volumes across the entire range $[a, b]$ gives the total volume, which is achieved through definite integration. The accuracy of the result depends heavily on the integration technique and the complexity of the function $f(x)$.
How to Calculate the Volume of a Solid of Revolution (Example)
Let’s calculate the volume of the solid formed by revolving the region under $f(x) = \sqrt{x}$ from $x=0$ to $x=4$ around the x-axis:
- **Identify the function and limits:** $f(x) = \sqrt{x}$, $a=0$, $b=4$.
- **Square the function:** Since the radius $r = f(x)$, the area of the disk is $A(x) = \pi [f(x)]^2 = \pi (\sqrt{x})^2 = \pi x$.
- **Set up the definite integral:** $V = \int_{0}^{4} A(x) dx = \pi \int_{0}^{4} x dx$.
- **Integrate:** The antiderivative of $x$ is $\frac{1}{2}x^2$.
- **Apply the limits:** $V = \pi \left[ \frac{1}{2}x^2 \right]_{0}^{4} = \pi \left( \frac{1}{2}(4)^2 – \frac{1}{2}(0)^2 \right)$.
- **Solve for V:** $V = \pi \left( \frac{16}{2} – 0 \right) = 8\pi$. The volume is approximately $25.1327$ cubic units.
Frequently Asked Questions (FAQ)
Is the Disk Method the only way to calculate the volume?
No. The two most common methods are the Disk/Washer Method and the Shell Method. The Disk Method is used when the slices are perpendicular to the axis of revolution, while the Shell Method is used when the slices are parallel to the axis of revolution.
What is the difference between Disk and Washer?
The Disk Method is a special case of the Washer Method. The Disk Method applies when the solid is adjacent to the axis of revolution (no hole in the center). The Washer Method applies when there is a hole, meaning the area is between two functions, $f(x)$ and $g(x)$.
Why does the formula use $f(x)^2$?
The cross-section of the solid is a circle (a disk). The area of a circle is $\pi r^2$. Since $f(x)$ represents the radius ($r$) of the disk at any given $x$, the area of the infinitesimal disk is $\pi [f(x)]^2$.
How accurate is this calculator?
Since this calculator uses numerical integration (specifically, the Trapezoidal Rule with a high number of steps), the result is a highly accurate approximation of the true volume. The accuracy is generally sufficient for engineering and academic purposes.