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Solid of Revolution — Definition, Formula & Examples

Solid of Revolution

A solid that is obtained by rotating a plane figure in space about an axis coplanar to the figure. The axis may not intersect the figure.

 

Two diagrams: a triangle in the xy-plane rotated about the y-axis produces a downward-pointing cone solid of revolution.

 

 

See also

Disk method, washer method, shell method, surface of revolution

Key Formula

V=π∫ab[f(x)]2 dxV = \pi \int_a^b \left[f(x)\right]^2 \, dx
Where:
  • VV = Volume of the solid of revolution
  • f(x)f(x) = The function being rotated around the x-axis
  • aa = Left endpoint of the interval of integration
  • bb = Right endpoint of the interval of integration
  • π\pi = Pi, approximately 3.14159, arising because each cross-section is a circle

Worked Example

Problem: Find the volume of the solid formed by rotating the region under y = x² from x = 0 to x = 3 about the x-axis.
Step 1: Identify the function and limits. The curve is f(x) = x² and the interval is [0, 3]. The rotation is about the x-axis, so use the disk method formula.
V=π∫03[x2]2 dxV = \pi \int_0^3 [x^2]^2 \, dx
Step 2: Simplify the integrand by squaring the function.
V=π∫03x4 dxV = \pi \int_0^3 x^4 \, dx
Step 3: Evaluate the integral using the power rule.
V=π[x55]03=π(355−055)V = \pi \left[\frac{x^5}{5}\right]_0^3 = \pi \left(\frac{3^5}{5} - \frac{0^5}{5}\right)
Step 4: Compute the numerical value.
V=π⋅2435=243π5V = \pi \cdot \frac{243}{5} = \frac{243\pi}{5}
Answer: The volume of the solid is 243π/5 ≈ 152.68 cubic units.

Another Example

This example differs by rotating about the y-axis instead of the x-axis, and uses the cylindrical shell method rather than the disk method. This shows how the choice of method depends on the axis of rotation and how the region is described.

Problem: Find the volume of the solid formed by rotating the region bounded by y = √x, y = 0, and x = 4 about the y-axis using the cylindrical shell method.
Step 1: Since we rotate about the y-axis but the function is given in terms of x, the shell method is convenient. A typical shell at position x has radius x, height √x, and thickness dx.
V=2π∫04x⋅x dxV = 2\pi \int_0^4 x \cdot \sqrt{x} \, dx
Step 2: Simplify the integrand by combining the powers of x.
V=2π∫04x3/2 dxV = 2\pi \int_0^4 x^{3/2} \, dx
Step 3: Integrate using the power rule.
V=2π[x5/25/2]04=2π[2x5/25]04V = 2\pi \left[\frac{x^{5/2}}{5/2}\right]_0^4 = 2\pi \left[\frac{2x^{5/2}}{5}\right]_0^4
Step 4: Evaluate at the bounds. Note that 4^(5/2) = (√4)^5 = 2^5 = 32.
V=2π⋅2(32)5=2π⋅645=128π5V = 2\pi \cdot \frac{2(32)}{5} = 2\pi \cdot \frac{64}{5} = \frac{128\pi}{5}
Answer: The volume of the solid is 128π/5 ≈ 80.42 cubic units.

Frequently Asked Questions

What is the difference between the disk method and the shell method for solids of revolution?
The disk (or washer) method slices the solid perpendicular to the axis of rotation, producing circular cross-sections whose areas you integrate. The shell method slices parallel to the axis of rotation, producing thin cylindrical shells whose lateral surface areas you integrate. Both give the same volume, but one is often algebraically simpler depending on whether the function is easier to express in terms of x or y.
How do you decide which axis to rotate around?
The axis of rotation is specified in the problem — you do not choose it yourself. However, the axis determines which method is most convenient. If you rotate around the x-axis and the curve is given as y = f(x), the disk method is usually straightforward. If you rotate around the y-axis with the same function, the shell method often avoids the need to solve for x in terms of y.
What is the washer method and when do you use it?
The washer method is a variation of the disk method used when the region being rotated does not touch the axis of rotation, creating a hole in the center of each cross-section. The formula is V = π∫[R(x)² − r(x)²] dx, where R(x) is the outer radius and r(x) is the inner radius. You use it whenever the solid has a hollow core.

Solid of Revolution vs. Surface of Revolution

Solid of RevolutionSurface of Revolution
What it isThe entire 3D volume enclosed by rotating a regionOnly the outer boundary surface formed by rotating a curve
What you calculateVolume (cubic units)Surface area (square units)
Key formula (about x-axis)V = π∫[f(x)]² dxS = 2π∫f(x)√(1 + [f'(x)]²) dx
ExampleThe volume of a sphere: 4πr³/3The surface area of a sphere: 4πr²

Why It Matters

Solids of revolution appear throughout AP Calculus AB/BC and university-level calculus courses as a central application of integration. Engineers use them to calculate volumes of tanks, pipes, nozzles, and any object with rotational symmetry. Understanding this concept also builds intuition for how one-dimensional integrals can measure three-dimensional quantities.

Common Mistakes

Mistake: Forgetting to square the function in the disk method.
Correction: Each cross-sectional disk has area πr², where r = f(x). You must integrate π[f(x)]², not πf(x). Omitting the square gives an incorrect, typically smaller, answer.
Mistake: Using the wrong method for the axis of rotation, leading to incorrect radius expressions.
Correction: When rotating about the y-axis with y = f(x), the shell radius is x (the distance from the y-axis), not f(x). Always sketch the region and identify which distance is the radius and which is the height before setting up the integral.

Related Terms