Worksheets · Calculus

Center of mass and centroids worksheet

A center of mass is a weighted average of position. For point masses the weights are the masses; for a rod they are the density along it, so the sums become integrals; for a flat region of even density, the centroid weighs every bit of area equally.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Center of Mass

Date Period

Find each center of mass or centroid. Give exact answers.

  1. Masses of 6,1 and 5 kg sit at (8,8),(−2,3) and (4,0). Find the center of mass.
  2. A rod lies on the x-axis from x=0 to x=3 (in meters) with density ρ(x)=3+2x kg/m. Find its mass and center of mass.
  3. Find the centroid of the region bounded by y=x3, the x-axis and x=1.
  4. Masses of 5,1 and 4 kg sit at (5,−2),(−4,−6) and (0,2). Find the center of mass.
  5. A rod lies on the x-axis from x=0 to x=7 (in meters) with density ρ(x)=2+x kg/m. Find its mass and center of mass.
  6. Find the centroid of the region bounded by y=4x, the x-axis and x=1.
  7. Masses of 7,5 and 7 kg sit at (8,−3),(5,−4) and (−2,0). Find the center of mass.
  8. A rod lies on the x-axis from x=0 to x=1 (in meters) with density ρ(x)=1+2x2 kg/m. Find its mass and center of mass.
  9. Find the centroid of the region bounded by y=1−x2 and the x-axis.
  10. Masses of 9,7 and 4 kg sit at (−2,4),(8,7) and (4,7). Find the center of mass.
  11. A rod lies on the x-axis from x=0 to x=5 (in meters) with density ρ(x)=1+x2 kg/m. Find its mass and center of mass.
  12. Find the centroid of the region bounded by y=2x3, the x-axis and x=3.
  13. Masses of 8,2 and 1 kg sit at (−4,0),(2,−2) and (7,−2). Find the center of mass.
  14. A rod lies on the x-axis from x=0 to x=7 (in meters) with density ρ(x)=6+3x kg/m. Find its mass and center of mass.
  15. Find the centroid of the region bounded by y=x, the x-axis and x=9.
  16. Masses of 6,6 and 9 kg sit at (−1,7),(−6,0) and (−3,−6). Find the center of mass.
  17. A rod lies on the x-axis from x=0 to x=2 (in meters) with density ρ(x)=5−x2 kg/m. Find its mass and center of mass.
  18. Find the centroid of the region bounded by y=2x2, the x-axis and x=2.
  19. Masses of 1,1 and 4 kg sit at (6,3),(−2,−1) and (0,−1). Find the center of mass.
  20. A rod lies on the x-axis from x=0 to x=1 (in meters) with density ρ(x)=3−2x2 kg/m. Find its mass and center of mass.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Center of MassAnswer keyVersion 1

Date Period

Find each center of mass or centroid. Give exact answers.

  1. Masses of 6,1 and 5 kg sit at (8,8),(−2,3) and (4,0). Find the center of mass.(112,174)
  2. A rod lies on the x-axis from x=0 to x=3 (in meters) with density ρ(x)=3+2x kg/m. Find its mass and center of mass.m=18 kgxˉ=74 m
  3. Find the centroid of the region bounded by y=x3, the x-axis and x=1.(45,27)
  4. Masses of 5,1 and 4 kg sit at (5,−2),(−4,−6) and (0,2). Find the center of mass.(2110,−45)
  5. A rod lies on the x-axis from x=0 to x=7 (in meters) with density ρ(x)=2+x kg/m. Find its mass and center of mass.m=772 kgxˉ=14033 m
  6. Find the centroid of the region bounded by y=4x, the x-axis and x=1.(35,32)
  7. Masses of 7,5 and 7 kg sit at (8,−3),(5,−4) and (−2,0). Find the center of mass.(6719,−4119)
  8. A rod lies on the x-axis from x=0 to x=1 (in meters) with density ρ(x)=1+2x2 kg/m. Find its mass and center of mass.m=53 kgxˉ=35 m
  9. Find the centroid of the region bounded by y=1−x2 and the x-axis.(0,25)
  10. Masses of 9,7 and 4 kg sit at (−2,4),(8,7) and (4,7). Find the center of mass.(2710,11320)
  11. A rod lies on the x-axis from x=0 to x=5 (in meters) with density ρ(x)=1+x2 kg/m. Find its mass and center of mass.m=1403 kgxˉ=405112 m
  12. Find the centroid of the region bounded by y=2x3, the x-axis and x=3.(125,1087)
  13. Masses of 8,2 and 1 kg sit at (−4,0),(2,−2) and (7,−2). Find the center of mass.(−2111,−611)
  14. A rod lies on the x-axis from x=0 to x=7 (in meters) with density ρ(x)=6+3x kg/m. Find its mass and center of mass.m=2312 kgxˉ=14033 m
  15. Find the centroid of the region bounded by y=x, the x-axis and x=9.(275,98)
  16. Masses of 6,6 and 9 kg sit at (−1,7),(−6,0) and (−3,−6). Find the center of mass.(−237,−47)
  17. A rod lies on the x-axis from x=0 to x=2 (in meters) with density ρ(x)=5−x2 kg/m. Find its mass and center of mass.m=223 kgxˉ=911 m
  18. Find the centroid of the region bounded by y=2x2, the x-axis and x=2.(32,125)
  19. Masses of 1,1 and 4 kg sit at (6,3),(−2,−1) and (0,−1). Find the center of mass.(23,−13)
  20. A rod lies on the x-axis from x=0 to x=1 (in meters) with density ρ(x)=3−2x2 kg/m. Find its mass and center of mass.m=73 kgxˉ=37 m

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the centroid of the region bounded by y=x2 and y=2x.

  1. The curves meet at x = 0 and x = 2. The area is ∫ from 0 to 2 of (2x - x²) dx = 4/3.
  2. x̄ = (1/A) ∫ x(2x - x²) dx = (3/4)(4/3) = 1.
  3. ȳ = (1/A) ∫ ((2x)² - (x²)²)/2 dx = (3/4)(32/15) = 8/5.

The answer is (1,85).

Where students go wrong

Forgetting the one half in ȳ. The moment about the x-axis uses the midpoint of each vertical strip, (f + g)/2, which is why the integrand is (f² - g²)/2 and not f² - g².

Also called centroids, center of gravity, moments and center of mass or centroid of a region.

What you can put on this worksheet

Point masses on a line or in the plane
Masses of 5,6 and 7 kg sit at x=−5,2 and 8. Find the center of mass. → xˉ=4318
A rod with variable density
A rod lies on the x-axis from x=0 to x=2 (in meters) with density ρ(x)=4+3x kg/m. Find its center of mass. → xˉ=87 m
The centroid of a region
Find the centroid of the region bounded by y=3x2, the x-axis and x=3. → (94,8110)

Questions about these worksheets

Yes. Every center of mass and centroids sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the center of mass and centroids questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: point masses on a line or in the plane, a rod with variable density and the centroid of a region. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting the one half in ȳ. The moment about the x-axis uses the midpoint of each vertical strip, (f + g)/2, which is why the integrand is (f² - g²)/2 and not f² - g².