Worksheet: Applications of Double Integrals: Center of Mass

In this worksheet, we will practice finding the center of mass of plane regions with constant or variable density using double integrals.

Q1:

Find the center of mass of the region 𝑅=(𝑥,𝑦)𝑦0,𝑥+𝑦𝑎: with the given density function 𝜌(𝑥,𝑦)=1.

  • A 4 𝑎 3 𝜋 , 0
  • B 0 , 2 𝑎 3 𝜋
  • C 2 𝑎 3 𝜋 , 0
  • D 4 𝑎 3 𝜋 , 4 𝑎 3 𝜋
  • E 0 , 4 𝑎 3 𝜋

Q2:

Find the center of mass of the region 𝑅=(𝑥,𝑦)𝑦0,𝑥+𝑦1: with the given density function 𝜌(𝑥,𝑦)=𝑦.

  • A 0 , 𝜋 2
  • B 0 , 3 𝜋 8
  • C 1 2 , 1 2
  • D 0 , 3 𝜋 1 6
  • E 0 , 1 6 3 𝜋

Q3:

Find the center of mass of the region 𝑅=(𝑥,𝑦)𝑦0,𝑥0,1𝑥+𝑦4: with the given density function 𝜌(𝑥,𝑦)=𝑥+𝑦.

  • A 7 𝜋 6 , 7 𝜋 6
  • B 0 , 7 𝜋 6
  • C 4 5 𝜋 1 4 , 0
  • D 0 , 4 5 𝜋 1 4
  • E 4 5 𝜋 1 4 , 4 5 𝜋 1 4

Q4:

Find the center of mass of the region 𝑅=(𝑥,𝑦)0𝑥1,0𝑦𝑥: with the given density function 𝜌(𝑥,𝑦)=𝑥+𝑦.

  • A 2 1 1 7 , 1 4 7 5 5
  • B 1 7 9 0 , 1 1 1 2 6
  • C 1 7 2 1 , 5 5 1 4 7
  • D 3 0 7 , 5 5 1 4 7
  • E 5 5 1 4 7 , 1 7 2 1

Q5:

Find the center of mass of the region 𝑅={(𝑥,𝑦)0𝑥2,0𝑦4}: with the given density function 𝜌(𝑥,𝑦)=2𝑦.

  • A 1 6 3 , 2
  • B 2 , 1 6 3
  • C 8 3 , 1
  • D 1 , 3 8
  • E 1 , 8 3

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