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.

  • A4𝑎3𝜋,0
  • B0,2𝑎3𝜋
  • C2𝑎3𝜋,0
  • D4𝑎3𝜋,4𝑎3𝜋
  • E0,4𝑎3𝜋

Q2:

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

  • A0,𝜋2
  • B0,3𝜋8
  • C12,12
  • D0,3𝜋16
  • E0,163𝜋

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.

  • A7𝜋6,7𝜋6
  • B0,7𝜋6
  • C45𝜋14,0
  • D0,45𝜋14
  • E45𝜋14,45𝜋14

Q4:

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

  • A2117,14755
  • B1790,11126
  • C1721,55147
  • D307,55147
  • E55147,1721

Q5:

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

  • A163,2
  • B2,163
  • C83,1
  • D1,38
  • E1,83

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