Centre of mass of a solid of revolution

The formula

For the uniform solid formed by rotating y = f(x) from x = a to x = b about the x-axis:

x̄ = ∫πy² x dx / ∫πy² dx.

The π cancels, but keep it in the volume if the question needs the mass. By symmetry ȳ = 0.

Proving the standard results

• Solid hemisphere: rotate y² = r² - x² from 0 to r. ∫(r² - x²)x dx = r⁴/4 and ∫(r² - x²) dx = 2r³/3, so x̄ = 3r/8.
• Solid cone with vertex at O: rotate y = (r/h)x from 0 to h. x̄ = (∫x³ dx)/(∫x² dx) = (h⁴/4)/(h³/3) = 3h/4 from the vertex, which is h/4 from the base.

Examiner advice

Use the letters the question uses. In June 2025 some candidates wrote r when the radius was given as a, and lost the final mark. If the question gives you a volume, use it rather than integrating again.

Worked example

Use integration to show that the centre of mass of a uniform solid hemisphere of radius r is 3r/8 from the centre of its plane face.

1. Rotate y² = r² - x² for 0 ≤ x ≤ r about the x-axis.
2. ∫π(r² - x²)x dx = π[r^2x2/2 - x⁴/4] = πr⁴/4.
3. ∫π(r² - x²) dx = π[r^2x - x³/3] = 2πr³/3.
4. x̄ = (πr⁴/4)/(2πr³/3) = 3r/8.

Answer: x̄ = 3r/8

Check yourself

1. The region under y = x² from x = 0 to x = 1 is rotated about the x-axis to form a uniform solid. Find x̄, to 3 significant figures.
   Show answer

   x̄ = 0.833. y² = x⁴. ∫x⁵ dx = 1/6, ∫x⁴ dx = 1/5, so x̄ = 5/6 = 0.833.

2. The region under y = sqrt x from x = 0 to x = 4 is rotated about the x-axis. Find x̄ for the uniform solid, to 3 significant figures.
   Show answer

   x̄ = 2.67. y² = x. ∫x² dx = 64/3, ∫x dx = 8, x̄ = 8/3 = 2.67.

3. Find the volume of the solid formed by rotating y = sqrt x, from x = 0 to x = 4, about the x-axis, to 3 significant figures.
   Show answer

   25.1. V = π∫x dx = 8π = 25.1.

4. The line y = x/2 from x = 0 to x = 8 is rotated about the x-axis to form a uniform solid cone with its vertex at O. Use integration to find x̄.
   Show answer

   x̄ = 6. x̄ = ∫x³ dx / ∫x² dx = (1024)/(512/3) = 6, so 6 from the vertex (2 from the base, which is h/4).

5. A uniform solid hemisphere has radius 5 cm. Find the distance of its centre of mass from the centre of the plane face, to 3 significant figures.
   Show answer

   1.88 cm. 3r/8 = 1.88 cm.

6. Which integral is the numerator when finding x̄ for a uniform solid of revolution about the x-axis?
   • ∫πy² dx
   • ∫πy² x dx
   • ∫xy dx
   • ∫½y² dx
   Show answer

   ∫πy² x dx. Each disc has volume πy² δx and sits at x.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M3
• Pearson Edexcel IAL Mechanics M3 (WME03) question paper, June 2025: front-cover instruction on g and accuracy
• Pearson Edexcel IAL Mechanics M3 (WME03) examiners' report, June 2025

Last updated 8 November 2026. Spotted a mistake? Tell us.