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 ) 0 𝑥 1 , 0 𝑦 1 , 0 𝑧 1 𝑥 𝑦 } : with the given density function 𝜌 ( 𝑥 , 𝑦 , 𝑧 ) = 1 .

  • A 1 6 , 1 6 , 1 6
  • B 1 2 , 1 2 , 1 2
  • C 1 4 , 1 4 , 1 4
  • D ( 4 , 4 , 4 )
  • E 1 8 , 1 8 , 1 8

Q2:

Find the center of mass of the solid 𝑆 = ( 𝑥 , 𝑦 , 𝑧 ) 𝑥 0 , 𝑦 0 , 𝑧 0 , 𝑥 + 𝑦 + 𝑧 𝑎 : with the given density function 𝜌 ( 𝑥 , 𝑦 , 𝑧 ) = 1 .

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

Q3:

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

  • A 1 3 , 1 3 , 1 3
  • B ( 1 , 1 , 1 )
  • C 2 3 , 2 3 , 2 3
  • D 3 2 , 3 2 , 3 2
  • E 1 2 , 1 2 , 1 2

Q4:

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

  • A 5 1 2 , 5 1 2 , 5 1 2
  • B 1 2 7 , 1 2 7 , 1 2 7
  • C 5 7 , 5 7 , 5 7
  • D ( 1 , 1 , 1 )
  • E 7 1 2 , 7 1 2 , 7 1 2

Q5:

Find the center of mass of the solid 𝑆 = ( 𝑥 , 𝑦 , 𝑧 ) 𝑧 0 , 𝑥 + 𝑦 + 𝑧 𝑎 : with the given density function 𝜌 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 𝑧 .

  • A 0 , 0 , 5 𝑎 8
  • B 0 , 0 , 5 𝑎 6
  • C 0 , 0 , 5 𝑎 1 2
  • D 0 , 0 , 5 𝑎 3
  • E 0 , 0 , 1 2 5 𝑎

Q6:

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.