This calculator determines the Volume of Revolution ($V$) generated by revolving the function $f(x) = A \cdot x^p$ around the x-axis, bounded by the integration limits $a$ and $b$.
Solids of Revolution Calculator
Volume of Revolution (V):
Solids of Revolution Volume Formula
Disk Method Formula (Volume $V$)
$$V = \pi \int_a^b [f(x)]^2 dx$$For $f(x) = A \cdot x^p$, the solved integral is:
$$V = \pi A^2 \left[ \frac{b^{2p+1}}{2p+1} – \frac{a^{2p+1}}{2p+1} \right]$$ Formula Source: Wikipedia – Solid of Revolution | Wolfram MathWorld ReferenceVariables Explained
- Function Coefficient ($A$): The constant scaling factor in the radius function $f(x)$.
- Function Exponent ($p$): The power to which the variable $x$ is raised in the function $f(x)$.
- Lower Limit of Integration ($a$): The starting point on the x-axis for the calculation.
- Upper Limit of Integration ($b$): The ending point on the x-axis for the calculation.
- Volume ($V$): The final calculated volume of the solid generated.
What is a Solid of Revolution?
A solid of revolution is a three-dimensional shape obtained by rotating a two-dimensional curve or planar region around a straight line (the axis of revolution). This concept is fundamental in integral calculus and is used to calculate volumes of complex, non-standard shapes. The resulting solid often possesses a high degree of symmetry around the axis of revolution.
The primary methods for calculating the volume of these solids are the Disk Method, the Washer Method (for regions that don’t touch the axis of revolution), and the Shell Method. This calculator specifically uses the Disk Method, which is suitable when the area being revolved is flush against the axis of rotation, generating a solid without a hole in the center.
How to Calculate Solids of Revolution (Example)
Using the function $f(x) = 2x^2$ (where $A=2, p=2$) revolved around the x-axis from $a=0$ to $b=1$.
- Define the Function and Limits: $A=2$, $p=2$, $a=0$, $b=1$.
- Apply the Formula: Substitute the values into the derived volume formula: $$V = \pi (2)^2 \left[ \frac{1^{2(2)+1}}{2(2)+1} – \frac{0^{2(2)+1}}{2(2)+1} \right]$$
- Simplify the Exponent: The exponent $2p+1$ becomes $2(2)+1 = 5$.
- Calculate the Integral: $$V = 4\pi \left[ \frac{1^5}{5} – \frac{0^5}{5} \right] = 4\pi \left[ \frac{1}{5} – 0 \right]$$
- Final Volume: $V = \frac{4\pi}{5} \approx 2.5133$.
Frequently Asked Questions (FAQ)
What is the difference between the Disk and Washer Methods?
The Disk Method is used when the area being rotated is flush against the axis of revolution. The Washer Method is used when there is a gap between the area and the axis, resulting in a solid with a hole (a ‘washer’ cross-section).
What happens if the exponent $p$ makes $2p+1$ equal to zero?
If $2p+1 = 0$, then $p = -1/2$. The formula used by this calculator is invalid for this case, as it results in division by zero. For $f(x) = A/\sqrt{x}$, the integral $\int x^{-1} dx = \ln|x|$ must be used instead, which the calculator will flag as an error.
Can this calculator solve for revolutions around the y-axis?
No, this specific version is configured for revolution around the x-axis using the Disk Method on a function of $x$. For the y-axis, the function must typically be expressed as $x=g(y)$, and the integral must be taken with respect to $y$.
Why is $\pi$ used in the formula?
The Disk Method works by summing the volumes of infinitesimally thin cylinders (disks). The volume of a single disk is $\pi r^2 h$, where $r=f(x)$ is the radius and $h=dx$ is the thickness. The integral sums these $\pi [f(x)]^2 dx$ volumes.