Worksheet: Applications of Triple Integrals

In this worksheet, we will practice using triple integrals in applications like the center of mass and moment of inertia of a solid with constant or variable density.

Q1:

Find the center of mass of the solid 𝑆={(𝑥,𝑦,𝑧)0𝑥1,0𝑦1,0𝑧1𝑥𝑦}: with the given density function 𝜌(𝑥,𝑦,𝑧)=1.

  • A14,14,14
  • B12,12,12
  • C18,18,18
  • D16,16,16
  • E(4,4,4)

Q2:

Find the center of mass of the solid 𝑆=(𝑥,𝑦,𝑧)𝑥0,𝑦0,𝑧0,𝑥+𝑦+𝑧𝑎: with the given density function 𝜌(𝑥,𝑦,𝑧)=1.

  • A3𝑎8,3𝑎8,3𝑎8
  • B3𝑎16,3𝑎16,3𝑎16
  • C83𝑎,83𝑎,83𝑎
  • D2𝑎3,2𝑎3,2𝑎3
  • E𝑎2,𝑎2,𝑎2

Q3:

Find the center of mass of the solid 𝑆={(𝑥,𝑦,𝑧)0𝑥1,0𝑦1,0𝑧1}: with the given density function 𝜌(𝑥,𝑦,𝑧)=𝑥𝑦𝑧.

  • A(1,1,1)
  • B12,12,12
  • C23,23,23
  • D32,32,32
  • E13,13,13

Q4:

Find the center of mass of the solid 𝑆={(𝑥,𝑦,𝑧)0𝑥1,0𝑦1,0𝑧1}: with the given density function 𝜌(𝑥,𝑦,𝑧)=𝑥+𝑦+𝑧.

  • A57,57,57
  • B512,512,512
  • C127,127,127
  • D(1,1,1)
  • E712,712,712

Q5:

Find the center of mass of the solid 𝑆=(𝑥,𝑦,𝑧)𝑧0,𝑥+𝑦+𝑧𝑎: with the given density function 𝜌(𝑥,𝑦,𝑧)=𝑥+𝑦+𝑧.

  • A0,0,5𝑎8
  • B0,0,5𝑎6
  • C0,0,5𝑎12
  • D0,0,125𝑎
  • E0,0,5𝑎3

Q6:

Let 𝑎, 𝑏, and 𝑐 be real numbers selected randomly from the interval (0,1). What is the probability that the equation 𝑎𝑥+𝑏𝑥+𝑐=0 has at least one real solution for 𝑥? Rounding the value to four decimal places.

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