All subjects · Mathematics · WME03 Mechanics 3

Centre of mass of a lamina by integration

Edexcel IAL Mathematics · WME03 Mechanics 3

Pearson's specification says you must use integration and/or symmetry to find centres of mass. For a uniform lamina bounded by a curve, split it into thin vertical strips and add up their moments.

Practise this in the app: instant marking and a fix for every mistake. Free, no email needed to start.

Practise this lesson

The formulae

For the uniform lamina between y = f(x), the x-axis, x = a and x = b (with A = ∫y dx its area):

All integrals are from a to b.

Why they work

A strip of width δx at x has area y δx, so its mass is proportional to y δx and it sits at distance x from the y-axis. Its own centre is halfway up, at height ½y, which gives the ½y2 in the second formula. Summing the moments and letting δx tend to 0 gives the integrals.

Tips

Find the area first: you need it for both coordinates. Check the answer is inside the region and on any line of symmetry.

Worked example

Find the centre of mass of the uniform lamina bounded by y = x2, the x-axis and x = 2.

  1. Area = ∫x2 dx from 0 to 2 = 8/3.
  2. ∫x · x2 dx = [x4/4] = 4, so x̄ = 4 ÷ 8/3 = 1.5.
  3. ∫½x4 dx = [x5/10] = 3.2, so ȳ = 3.2 ÷ 8/3 = 1.2.

Answer: (1.5, 1.2)

Key idea: x̄ = ∫xy dx/∫y dx; ȳ = ∫½y2 dx/∫y dx.

Check yourself

  1. A uniform lamina is bounded by y = x2, the x-axis and x = 3. Find x̄.
    Show answer

    x̄ = 2.25. Area = 9, ∫x3 dx = 81/4, so x̄ = 2.25.

  2. For the same lamina (y = x2, x from 0 to 3), find ȳ.
    Show answer

    ȳ = 2.7. ∫½x4 dx = 243/10 = 24.3, so ȳ = 24.3/9 = 2.7.

  3. A uniform lamina is bounded by y = sqrt x, the x-axis and x = 4. Find x̄.
    Show answer

    x̄ = 2.4. Area = (2/3)(8) = 16/3. ∫x3/2 dx = (2/5)(32) = 12.8. x̄ = 12.8 ÷ 16/3 = 2.4.

  4. For the same lamina (y = sqrt x, x from 0 to 4), find ȳ.
    Show answer

    ȳ = 0.75. ∫½x dx = [x2/4] = 4, so ȳ = 4 ÷ 16/3 = 0.75.

  5. A uniform triangular lamina is bounded by y = x, the x-axis and x = 3. Use integration to find x̄.
    Show answer

    x̄ = 2. Area = 4.5, ∫x2 dx = 9, x̄ = 2. (This agrees with two thirds of the way along the median.)

  6. A uniform lamina is bounded by y = 4 - x2, the x-axis and the y-axis, for x from 0 to 2. Find x̄.
    Show answer

    x̄ = 0.75. Area = 8 - 8/3 = 16/3. ∫x(4 - x2) dx = 8 - 4 = 4. x̄ = 4 ÷ 16/3 = 0.75.

  7. Which integral is the numerator for ȳ of a uniform lamina under y = f(x)?
    • ∫xy dx
    • ∫½y2 dx
    • ∫y2 dx
    • ∫½y dx
    Show answer

    ∫½y2 dx. Each strip has its centre at height ½y and area y δx.

Practise with instant feedback

Sources

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