Find The Volume Of The Resulting Solid Calculator

Calculate Volume by Disk Method

This calculator finds the volume of the solid generated by rotating the region under the line y = mx + c between x=a and x=b around the x-axis.

Results:

The volume (V) is calculated using the Disk Method for y=f(x)=mx+c rotated around the x-axis:
V = π ∫_a^b [f(x)]² dx = π ∫_a^b (mx+c)² dx
V = π [ (m²x³/3 + mcx² + c²x) |_a^b ]

Summary Table

Parameter	Value
Slope (m)
Y-Intercept (c)
Lower Limit (a)
Upper Limit (b)
f(a)
f(b)
Volume

Summary of inputs 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 shape or region around an axis. In calculus, this is a common application of integration. Our calculator specifically focuses on the solid generated by rotating the area under a straight line y=mx+c between x=a and x=b around the x-axis, using the Disk Method.

This type of calculator is useful for students learning calculus, engineers, physicists, and anyone needing to find the volume of such solids without performing the manual integration every time. The volume of solid of revolution calculator automates the integration process for a given function and limits.

Common misconceptions include thinking it applies to any solid or any axis of rotation with a single formula. However, different methods (Disk, Washer, Shell) and formulas are used depending on the shape of the region and the axis of rotation. This volume of solid of revolution calculator is specifically for the Disk Method around the x-axis for a linear function.

Volume of Solid of Revolution Formula and Mathematical Explanation

When a region bounded by a function y=f(x), the x-axis, and the lines x=a and x=b is rotated around the x-axis, the volume of the resulting solid can be found using the Disk Method. The idea is to slice the solid into infinitesimally thin disks perpendicular to the x-axis. Each disk at x has a radius f(x) and thickness dx, so its volume is dV = π[f(x)]²dx.

The total volume is the sum (integral) of the volumes of these disks from a to b:

V = π ∫_a^b [f(x)]² dx

For our specific volume of solid of revolution calculator, the function is linear: f(x) = mx + c. So, [f(x)]² = (mx + c)² = m²x² + 2mcx + c².

The integral becomes:

∫ (m²x² + 2mcx + c²) dx = m²x³/3 + mcx² + c²x

Evaluating this from a to b:

V = π [ (m²b³/3 + mcb² + c²b) – (m²a³/3 + mca² + c²a) ]

This is the formula our volume of solid of revolution calculator uses.

Variable	Meaning	Unit	Typical Range
m	Slope of the line f(x)=mx+c	Dimensionless (if x,y are lengths)	Any real number
c	Y-intercept of the line f(x)=mx+c	Same as y	Any real number
a	Lower limit of integration	Same as x	a < b
b	Upper limit of integration	Same as x	b > a
V	Volume of the solid	Cubic units	≥ 0

Variables used in the volume calculation.

Practical Examples (Real-World Use Cases)

Let's see how our volume of solid of revolution calculator can be used.

Example 1: Volume of a Cone

A cone with height H and base radius R can be generated by rotating the line y = (R/H)x from x=0 to x=H around the x-axis. Here, m = R/H, c = 0, a = 0, b = H.

If H=6 and R=3, then m=3/6=0.5, c=0, a=0, b=6.

Using the calculator with m=0.5, c=0, a=0, b=6: Volume = π * [ (0.5²*6³/3 + 0*0.5*6² + 0²*6) – (0) ] = π * [0.25 * 216 / 3] = π * [18] = 18π ≈ 56.55 cubic units. The formula for a cone's volume is (1/3)πR²H = (1/3)π(3²)(6) = 18π. It matches!

Example 2: Volume of a Cylinder

A cylinder with radius R and height H can be generated by rotating the line y=R from x=0 to x=H around the x-axis. Here, m=0, c=R, a=0, b=H.

If R=4 and H=5, then m=0, c=4, a=0, b=5.

Using the volume of solid of revolution calculator with m=0, c=4, a=0, b=5: Volume = π * [ (0 + 0 + 4²*5) – (0) ] = π * [16*5] = 80π ≈ 251.33 cubic units. The formula for a cylinder's volume is πR²H = π(4²)(5) = 80π. It matches!

How to Use This Volume of Solid of Revolution Calculator

Using our volume of solid of revolution calculator is straightforward:

1. Enter the Slope (m): Input the slope of the line y=mx+c that bounds the region.
2. Enter the Y-Intercept (c): Input the y-intercept of the line.
3. Enter the Lower Limit (a): Input the starting x-value of the region.
4. Enter the Upper Limit (b): Input the ending x-value of the region. Ensure b > a.
5. Calculate: Click the "Calculate Volume" button or just change the input values. The results will update automatically.
6. Read the Results: The primary result is the volume. Intermediate values like f(a) and f(b) are also shown.
7. Reset (Optional): Click "Reset" to return to default values.
8. Copy (Optional): Click "Copy Results" to copy the main volume and key parameters.

The calculator assumes rotation around the x-axis and uses the Disk Method for the function y=mx+c. Make sure your problem fits this setup.

Key Factors That Affect Volume Results

The volume calculated by the volume of solid of revolution calculator depends on several factors:

• The Function (m and c): The slope 'm' and y-intercept 'c' define the radius f(x) of the disks at each x. Larger f(x) values (due to larger m or c within the interval [a,b] if f(x)>0) lead to larger volumes.
• The Interval [a, b]: The width of the interval (b-a) directly affects the volume. A wider interval generally means more disks are being summed, increasing the volume.
• The Square of the Function: The volume depends on the integral of [f(x)]², so the magnitude of f(x) is very influential.
• Axis of Rotation: This calculator assumes rotation around the x-axis. Rotating around a different axis (like the y-axis or another line) would require a different setup (like the Shell Method or Washer Method if there's a hole) and result in a different volume. Our washer method volume calculator handles some variations.
• The Method Used: We use the Disk Method. If the region was bounded by two functions, or rotated around the y-axis, the Washer or Shell methods might be needed, yielding different volumes. Our shell method volume tool is useful here.
• Units: The units of the volume will be the cubic units of the x and y axes. If x and y are in cm, the volume is in cm³.

Frequently Asked Questions (FAQ)

Q: What if my function is not y=mx+c? A: This specific volume of solid of revolution calculator is designed for linear functions y=mx+c. For other functions, you would need to integrate [f(x)]² manually or use a more general integral calculator online with f(x)² as the integrand.

Q: How do I find the volume if I rotate around the y-axis? A: If you rotate around the y-axis, you'd typically use the Shell Method or integrate with respect to y. The formula changes. You might need to express x as a function of y.

Q: What is the difference between the Disk and Washer methods? A: The Disk Method is used when the region being rotated is flush against the axis of rotation. The Washer Method is used when there's a gap between the region and the axis, or when rotating the area between two curves, creating a solid with a hole. See our washer method volume page for more.

Q: Can 'a' be greater than 'b'? A: No, for standard integration from left to right, the lower limit 'a' must be less than the upper limit 'b'. The calculator enforces b > a.

Q: What if f(x) is negative in the interval [a,b]? A: The formula squares f(x), so [f(x)]² is always non-negative. The volume calculation still works, representing the volume of the solid formed by rotating the area between y=f(x) and the x-axis.

Q: How does this relate to finding the area under a curve? A: Finding the area under y=f(x) involves integrating f(x) dx. Finding the volume of revolution involves integrating π[f(x)]² dx. It's like summing areas of circles (πr²) instead of heights (f(x)). Check our area under a curve calculator.

Q: Can I use this calculator for a parabola? A: No, this volume of solid of revolution calculator is only for y=mx+c. For a parabola (e.g., y=ax²+bx+d), you'd need to integrate π(ax²+bx+d)² dx.

Q: Are there examples of solids of revolution? A: Yes, spheres, cones, cylinders, and frustums are common solids of revolution examples. Many machine parts also have shapes that are solids of revolution.