Find The Volume Of The Solid Generated By Revolving Calculator

This calculator finds the volume of a solid generated by revolving a region bounded by two functions (polynomials up to degree 2) around the x-axis (y=0) using the Washer Method (or Disk Method if the inner radius is zero).

Calculator

Outer Radius R(x) = Ax² + Bx + C

Inner Radius r(x) = Dx² + Ex + F

For the Disk Method, set D, E, and F to 0.

Integration Limits

Term	Coefficient in R(x)²	Coefficient in r(x)²
x⁴	0	0
x³	0	0
x²	0	0
x	0	0
const	0	0

Coefficients of the expanded forms of R(x)² and r(x)².

What is the Volume of a Solid of Revolution?

The volume of a solid of revolution is the volume of a three-dimensional object obtained by rotating a two-dimensional plane region around an axis (the axis of revolution) that lies in the same plane. Imagine taking a flat shape and spinning it around a line; the space it sweeps out forms the solid of revolution. Our volume of solid of revolution calculator helps you find this volume when the region is bounded by functions and revolved around the x-axis.

This concept is fundamental in calculus, particularly in integral calculus, as it provides a way to calculate the volumes of various shapes like spheres, cones, cylinders, and more complex forms by integrating the area of cross-sections.

Who should use it? Students learning integral calculus, engineers, physicists, and anyone needing to calculate volumes of rotationally symmetric objects will find a volume of solid of revolution calculator useful.

Common misconceptions include thinking the formula is the same for revolution around the x-axis and y-axis (it changes based on the axis and integration variable), or that the disk and washer methods are entirely different (the disk method is a special case of the washer method where the inner radius is zero).

Volume of Solid of Revolution Formula and Mathematical Explanation

When revolving a region bounded by y = R(x) (outer radius), y = r(x) (inner radius), x = a, and x = b around the x-axis (y=0), where R(x) ≥ r(x) ≥ 0 for all x in [a, b], we use the Washer Method. If r(x) = 0, it simplifies to the Disk Method.

The region is between the curves y = R(x) and y = r(x) from x = a to x = b. When we revolve this region around the x-axis, a typical cross-section at a given x, perpendicular to the axis of revolution, is a washer (or a disk if r(x)=0) with outer radius R(x) and inner radius r(x).

The area of this washer is A(x) = π(R(x)²) – π(r(x)²) = π(R(x)² – r(x)²).

To find the total volume, we integrate this cross-sectional area from x = a to x = b:

V = ∫[from a to b] A(x) dx = ∫[from a to b] π(R(x)² – r(x)²) dx = π ∫[from a to b] (R(x)² – r(x)²) dx

Our volume of solid of revolution calculator specifically handles cases where R(x) and r(x) are polynomials of degree up to 2: R(x) = Ax² + Bx + C and r(x) = Dx² + Ex + F.

So, we calculate π ∫[from a to b] ((Ax² + Bx + C)² – (Dx² + Ex + F)²) dx.

Variable	Meaning	Unit	Typical Range
R(x)	Outer radius function (distance from axis of revolution to outer curve)	Length	Depends on the function
r(x)	Inner radius function (distance from axis of revolution to inner curve)	Length	0 ≤ r(x) ≤ R(x)
a	Lower limit of integration along the x-axis	Length	Any real number
b	Upper limit of integration along the x-axis	Length	b ≥ a
V	Volume of the solid of revolution	Volume (Length³)	≥ 0
A, B, C	Coefficients of the outer radius polynomial R(x)	Varies	Any real numbers
D, E, F	Coefficients of the inner radius polynomial r(x)	Varies	Any real numbers

Variables used in the volume calculation.

Practical Examples (Real-World Use Cases)

Example 1: Volume of a Paraboloid

Find the volume of the solid generated by revolving the region bounded by y = x², x = 0, and x = 2 around the x-axis.

• Outer radius R(x) = x² (so A=1, B=0, C=0)
• Inner radius r(x) = 0 (so D=0, E=0, F=0) – This is the Disk Method.
• Lower limit a = 0
• Upper limit b = 2

Using the volume of solid of revolution calculator with these inputs: A=1, B=0, C=0, D=0, E=0, F=0, a=0, b=2.

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 ≈ 20.106 cubic units.

Example 2: Volume of a Solid with a Hole

Find the volume of the solid generated by revolving the region bounded by y = √x and y = x² between x=0 and x=1 around the x-axis.

For x between 0 and 1, √x ≥ x², so R(x) = √x and r(x) = x². However, our calculator takes polynomial inputs up to degree 2. Let’s modify: revolve the region between y=x (R(x)) and y=x² (r(x)) from x=0 to x=1.

• Outer radius R(x) = x (A=0, B=1, C=0)
• Inner radius r(x) = x² (D=1, E=0, F=0)
• Lower limit a = 0
• Upper limit b = 1

Using the volume of solid of revolution calculator with A=0, B=1, C=0, D=1, E=0, F=0, a=0, b=1.

V = π ∫[0 to 1] (x² – (x²)²) dx = π ∫[0 to 1] (x² – x⁴) dx = π [x³/3 – x⁵/5] from 0 to 1 = π (1/3 – 1/5) = π (2/15) ≈ 0.419 cubic units.

How to Use This Volume of Solid of Revolution Calculator

1. Identify Functions: Determine the outer radius function R(x) and inner radius function r(x) for the region being revolved around the x-axis. Ensure they are polynomials of degree 2 or less (Ax² + Bx + C). If the degree is lower, set the higher-order coefficients to zero.
2. Enter Coefficients for R(x): Input the values for A, B, and C for your outer radius function R(x).
3. Enter Coefficients for r(x): Input the values for D, E, and F for your inner radius function r(x). If using the Disk Method, set D=0, E=0, F=0.
4. Set Integration Limits: Enter the lower limit ‘a’ and upper limit ‘b’ of integration along the x-axis that define your region.
5. Calculate: The calculator automatically updates the Volume, intermediate integrals, table, and chart as you input values. You can also click “Calculate Volume”.
6. Read Results: The primary result is the Volume. Intermediate results show the integral of R(x)² and r(x)² over the limits.
7. Visualize: The chart plots R(x) and r(x) to help you see the region. The table shows coefficients after squaring R(x) and r(x).
8. Reset: Use the “Reset” button to return to default values.

This volume of solid of revolution calculator is designed for revolutions around the x-axis (y=0) and polynomial functions up to degree 2.

Key Factors That Affect Volume of Solid of Revolution Results

• Outer Radius Function R(x): The shape and magnitude of R(x) directly influence the outer boundary of the solid. Larger R(x) values lead to a larger volume.
• Inner Radius Function r(x): If r(x) > 0, it creates a hole in the solid. The larger r(x), the larger the hole and the smaller the volume of the solid material.
• Difference R(x)² – r(x)²: The volume is directly proportional to the integral of this difference. The greater the area between R(x)² and r(x)² over the interval [a, b], the larger the volume.
• Limits of Integration (a and b): The interval [a, b] determines the length along the x-axis over which the region is defined and revolved. A wider interval generally leads to a larger volume, assuming R(x)² – r(x)² is positive.
• Axis of Revolution: This calculator assumes revolution around the x-axis (y=0). Revolving around a different axis (e.g., y=k, x=h, or y-axis) would require a different setup and formula, often involving shifting the functions or using the cylindrical shells method (see Cylindrical Shells Method).
• The Functions’ Behavior: Whether the functions are increasing or decreasing, and where they intersect, defines the region and thus the solid’s shape and volume.

Frequently Asked Questions (FAQ)

1. What is the difference between the Disk and Washer methods?

The Disk Method is used when the region being revolved is bounded by one function and the axis of revolution, forming a solid without a hole. The Washer Method is used when the region is between two functions, creating a solid with a hole. The Disk Method is a special case of the Washer Method where the inner radius r(x) is 0. Our volume of solid of revolution calculator handles both.

2. Can this calculator handle revolution around the y-axis?

No, this specific calculator is set up for revolution around the x-axis (y=0). For revolution around the y-axis, you would need to express your functions as x in terms of y (x=g(y)) and integrate with respect to y, or use the Cylindrical Shells method. See our guide on Volume around y-axis.

3. What if my functions are not polynomials of degree 2 or less?

This calculator is limited to R(x) = Ax²+Bx+C and r(x) = Dx²+Ex+F. For other functions (e.g., trigonometric, exponential, or higher-degree polynomials), the integration of R(x)² and r(x)² would be different, and you’d need a more advanced integration tool or manual calculation.

4. How do I find the volume if the region is revolved around a line y=k (not the x-axis)?

If revolving around y=k, the radii become |R(x)-k| and |r(x)-k|. You would need to adjust the functions R(x) and r(x) before using the formula, ensuring you correctly identify the new outer and inner radii relative to y=k.

5. What if R(x) < r(x) in some parts of the interval?

The formula V = π ∫ (R(x)² – r(x)²) dx assumes R(x) ≥ r(x) ≥ 0 (or |R(x)| ≥ |r(x)| if around an axis other than y=0 and functions go below). If R(x) < r(x), you've likely misidentified the outer and inner radii. The outer radius is always the one further from the axis of revolution.

6. What units will the volume be in?

The units of the volume will be the cube of the units used for x, R(x), and r(x). If x is in cm, the volume is in cm³.

7. How does the chart help?

The chart visually represents the functions R(x) and r(x) over the interval [a, b], allowing you to see the 2D region that is being revolved around the x-axis.

8. Can I use this for the Cylindrical Shells method?

No, this calculator is based on the Disk/Washer method. The Cylindrical Shells method uses a different formula (V = 2π ∫ [a to b] x * (height) dx for revolution around y-axis, or 2π ∫ [c to d] y * (width) dy for revolution around x-axis). We have a separate Cylindrical Shells Calculator.

Related Tools and Internal Resources

• Area Between Curves Calculator: Calculates the area between two curves, which forms the basis of the region being revolved.
• Cylindrical Shells Method Calculator: Calculates the volume of solids of revolution using the cylindrical shells method, often useful for revolutions around the y-axis.
• Definite Integral Calculator: A general tool to calculate definite integrals of various functions, which can be used if R(x)² – r(x)² is known and integrable.
• Understanding Solids of Revolution: A guide explaining the concepts behind volumes of solids of revolution and different methods.
• Arc Length Calculator: Calculates the length of a curve defined by a function.
• Washer Method vs Shell Method: A comparison of the two main methods for finding volumes of revolution.