Find The Volume Of The Solid Formed By Rotating Calculator

Calculate the volume of a solid formed by rotating a curve y = f(x) = ax² + bx + c around the x-axis between x=x₁ and x=x₂ using our volume of solid of revolution calculator. Instantly find the volume using the disk method.

Calculator Inputs

Enter the coefficients of the quadratic function y = ax² + bx + c and the limits of integration.

Summary Table

Parameter	Value
Coefficient a	0
Coefficient b	1
Coefficient c	0
Lower Limit x₁	0
Upper Limit x₂	1
Volume	0.00

Input values and calculated volume.

What is a Volume of Solid of Revolution Calculator?

A volume of solid of revolution calculator is a tool used to determine the volume of a three-dimensional object formed by rotating a two-dimensional curve around an axis. Typically, this involves rotating a function y=f(x) around the x-axis or y-axis, or another line, over a specified interval. This calculator specifically uses the disk method for rotating y = ax² + bx + c around the x-axis.

Students of calculus, engineers, physicists, and mathematicians often use such calculators to find volumes of shapes that are otherwise difficult to calculate using simple geometric formulas. Common misconceptions include thinking it only applies to simple shapes or that it always involves rotation around the x or y-axis only (it can be around any line).

Volume of Solid of Revolution Formula and Mathematical Explanation (Disk Method)

When we rotate a continuous non-negative function y = f(x) around the x-axis from x = a to x = b, we can imagine slicing the resulting solid into an infinite number of thin disks perpendicular to the x-axis. Each disk has a radius r = f(x) and an infinitesimal thickness dx. The volume of one such disk is dV = π * (radius)² * thickness = π * (f(x))² dx.

To find the total volume, we integrate these infinitesimally thin disk volumes from x = a to x = b:

V = ∫[a to b] π * (f(x))² dx = π * ∫[a to b] (f(x))² dx

In our calculator, f(x) = ax² + bx + c. So we need to integrate:

(f(x))² = (ax² + bx + c)² = a²x⁴ + 2abx³ + (b²+2ac)x² + 2bcx + c²

The integral (antiderivative F(x)) is:

F(x) = (a²/5)x⁵ + (ab/2)x⁴ + ((b²+2ac)/3)x³ + bcx² + c²x

The volume is then V = π * (F(b) – F(a)), where a=x₁ and b=x₂.

Variables Table

Variable	Meaning	Unit	Typical Range
a, b, c	Coefficients of f(x) = ax² + bx + c	Dimensionless	Any real number
x₁, x₂	Limits of integration (start and end x-values)	Length units	Any real numbers, typically x₁ ≤ x₂
f(x)	The function being rotated	Length units	Depends on x, a, b, c
V	Volume of the solid of revolution	Cubic length units	≥ 0

Practical Examples (Real-World Use Cases)

Example 1: Volume of a Paraboloid

Let's find the volume of the solid generated by rotating the parabola y = x² (so a=1, b=0, c=0) around the x-axis from x=0 to x=2.

• a = 1, b = 0, c = 0
• x₁ = 0, x₂ = 2

Using the volume of solid of revolution calculator with these inputs, we get V = π * ∫[0 to 2] (x²)² dx = π * ∫[0 to 2] x⁴ dx = π * [x⁵/5] from 0 to 2 = π * (32/5 – 0) = 32π/5 ≈ 20.11 cubic units.

Example 2: Volume of a Truncated Cone-like Shape

Consider rotating the line y = 2x + 1 (a=0, b=2, c=1) around the x-axis from x=1 to x=3.

• a = 0, b = 2, c = 1
• x₁ = 1, x₂ = 3

V = π * ∫[1 to 3] (2x + 1)² dx = π * ∫[1 to 3] (4x² + 4x + 1) dx = π * [4x³/3 + 2x² + x] from 1 to 3 = π * [(36 + 18 + 3) – (4/3 + 2 + 1)] = π * [57 – 10/3] = π * (171-10)/3 = 161π/3 ≈ 168.60 cubic units. Our solid of revolution volume calculator can quickly compute this.

How to Use This Volume of Solid of Revolution Calculator

1. Enter Coefficients: Input the values for 'a', 'b', and 'c' for your quadratic function y = ax² + bx + c. If your function is linear (like y=mx+c), set 'a' to 0. If it's a constant (y=c), set 'a' and 'b' to 0.
2. Enter Limits: Input the lower limit x₁ and the upper limit x₂ for the integration. These define the portion of the curve being rotated.
3. Calculate: The calculator automatically updates the volume and intermediate results as you type. You can also click "Calculate Volume".
4. Read Results: The primary result is the volume. Intermediate values show the antiderivative evaluated at the limits.
5. Visualize: The chart shows the graph of your function between the limits.
6. Reset: Use the "Reset" button to go back to default values.
7. Copy: Use "Copy Results" to copy the main volume and inputs to your clipboard.

Decision-making: The calculated volume can be used in engineering design, physics problems (like finding the mass of an object with variable density), or simply for understanding calculus concepts. Ensure your limits x₁ and x₂ are correctly entered and that x₁ ≤ x₂ for standard integration.

Key Factors That Affect Volume of Solid of Revolution Results

• The Function f(x): The shape of the curve defined by f(x) (determined by a, b, c) directly impacts the radius of the disks at each x, and thus the volume. Larger f(x) values generally lead to larger volumes.
• Limits of Integration (x₁, x₂): The interval [x₁, x₂] determines the length along the x-axis over which the solid is generated. A wider interval generally results in a larger volume.
• The Axis of Rotation: Our calculator assumes rotation around the x-axis (y=0). Rotating around a different axis (e.g., y=k or the y-axis using the shell method) would yield a different volume and require a different formula (see our washer method volume calculator for rotation around y=k).
• Square of the Function: The volume depends on the integral of (f(x))², so the magnitude of f(x) is squared, making the volume sensitive to the height of the function from the axis.
• Continuity of f(x): The disk method assumes f(x) is continuous over [x₁, x₂]. Discontinuities would require separate integrals.
• Whether f(x) is Non-negative: If f(x) is negative, (f(x))² is still positive. The formula calculates the volume generated by |f(x)|. If you rotate the area between two curves, you'd use the washer method.

Understanding these factors helps in predicting how changes in the input function or limits will affect the final volume calculated by the volume of solid of revolution calculator.

Frequently Asked Questions (FAQ)

What if my function is not a quadratic?

This specific calculator is designed for f(x) = ax² + bx + c. For other functions, you would need to integrate (f(x))² manually or use a more advanced integral calculator volume tool that accepts general functions.

What if I rotate around the y-axis?

Rotation around the y-axis requires either expressing x as a function of y and using the disk method with dy, or using the shell method with dx. This calculator does not handle y-axis rotation directly.

What if the curve goes below the x-axis?

Since the formula uses (f(x))², the result will be the same as if you used |f(x)|. The volume is calculated based on the distance from the x-axis, squared.

Can I use this for the washer method?

No, this is specifically for the disk method (rotating the area between y=f(x) and the x-axis). For the washer method (area between two curves f(x) and g(x)), you'd integrate π((Outer Radius)² – (Inner Radius)²). See our washer method volume page.

What are the units of the volume?

The units of the volume will be the cubic units of whatever length units were used for x and f(x). If x is in cm, the volume is in cm³.

What if x₁ > x₂?

The integral from a to b is the negative of the integral from b to a. However, volume should be positive. It's conventional to set x₁ as the lower limit and x₂ as the upper limit (x₁ ≤ x₂). Our calculator will proceed but the geometric interpretation assumes x₁ ≤ x₂.

How accurate is this volume of solid of revolution calculator?

The calculation is based on the exact integral formula and should be very accurate, limited only by the precision of JavaScript's floating-point numbers.

Can I find the volume for trigonometric or exponential functions?

Not with this calculator directly, as it's for polynomials up to degree 2. You'd need a tool that can integrate (sin(x))², (e^x)², etc., or our definite integral calculator.