Worksheet: Triple Integrals

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In this worksheet, we will practice solving triple integration problems as well as applications of triple integration.

Q1:

Evaluate the triple integral 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 3 0 2 0 1 0 d d d .

Q2:

Evaluate the triple integral 1 𝑥 𝑦 𝑧 2 1 4 2 3 0 d d d .

Q3:

Evaluate the triple integral 𝑥 𝑦 𝑧 𝑧 𝑦 𝑥 1 0 𝑥 0 𝑦 0 d d d .

  • A 1 1 2
  • B 1 2 4
  • C 1 6
  • D 1 4 8
  • E 1 3 0

Q4:

Evaluate the triple integral 𝑧 𝑥 𝑥 𝑧 𝑦 𝑒 1 𝑦 0 0 2 1 𝑦 d d d .

  • A1
  • B 1 2
  • C 1 3
  • D 1 6
  • E − 5 6

Q5:

Evaluate the triple integral 𝑦 𝑧 𝑥 𝑧 𝑦 2 1 𝑦 0 𝑧 0 2 2 d d d .

  • A 1 0 2 3
  • B 1 2 8 5
  • C 5 1 2 2 0
  • D 1 0 2 3 4 0
  • E 3 4 1 9

Q6:

Evaluate the triple integral 1 𝑧 𝑦 𝑥 1 0 1 − 𝑥 0 1 − 𝑥 − 𝑦 0 d d d .

  • A 1 3
  • B − 1 2
  • C 1 2
  • D 1 6
  • E 1 4

Q7:

Evaluate the triple integral 𝑧 𝑒 𝑥 𝑦 𝑧 1 0 𝑧 0 𝑦 0 𝑦 2 d d d .

  • A 𝑒 4
  • B 1 2 ( 𝑒 − 2 )
  • C ( 𝑒 − 2 )
  • D 1 4 ( 𝑒 − 2 )
  • E ( 𝑒 − 4 )

Q8:

Find the centre 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 2 , 1 2 , 1 2
  • B 1 8 , 1 8 , 1 8
  • C 1 6 , 1 6 , 1 6
  • D 1 4 , 1 4 , 1 4
  • E ( 4 , 4 , 4 )

Q9:

Find the centre of mass of the solid 𝑆 = ( 𝑥 , 𝑦 , 𝑧 ) 𝑥 ≥ 0 , 𝑦 ≥ 0 , 𝑧 ≥ 0 , 𝑥 + 𝑦 + 𝑧 ≤ 𝑎 : 2 2 2 2 with the given density function 𝜌 ( 𝑥 , 𝑦 , 𝑧 ) = 1 .

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

Q10:

Find the centre 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 , 1 , 1 )
  • B 1 3 , 1 3 , 1 3
  • C 1 2 , 1 2 , 1 2
  • D 2 3 , 2 3 , 2 3
  • E 3 2 , 3 2 , 3 2

Q11:

Find the centre 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 2 2 2 .

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

Q12:

Find the centre of mass of the solid 𝑆 = ( 𝑥 , 𝑦 , 𝑧 ) 𝑧 ≥ 0 , 𝑥 + 𝑦 + 𝑧 ≤ 𝑎 : 2 2 2 2 with the given density function 𝜌 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 𝑧 2 2 2 .

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

Q13:

Evaluate the triple integral 𝑧 𝑥 𝑥 𝑦 𝑧 1 0 𝑧 0 𝑦 0 2 d d d .

  • A1
  • B 1 2
  • C 1 3
  • D 1 7 2
  • E 1 1 2

Q14:

Find the volume inside the cone 𝑧 = √ 𝑥 + 𝑦 2 2 , where 0 ≤ 𝑧 ≤ 3 .

  • A 9 − 2 √ 3 𝜋
  • B 1 8 𝜋
  • C √ 3 𝜋
  • D 9 𝜋
  • E 9 2

Q15:

Find the volume inside the cone 𝑧 = 𝑥 + 𝑦 2 2 , where 0 ≤ 𝑧 ≤ 4 .

  • A 4 𝜋
  • B 1 6 𝜋
  • C 6 𝜋
  • D 8 𝜋
  • E 3 𝜋

Q16:

Find the volume inside the elliptic cylinder 𝑥 𝑎 + 𝑦 𝑏 = 1 2 2 2 2 , where 0 ≤ 𝑧 ≤ 2 .

  • A 𝜋 𝑎 𝑏
  • B 4 𝜋 𝑎 𝑏
  • C 𝜋 ( 𝑎 𝑏 ) 2
  • D 2 𝜋 𝑎 𝑏
  • E 4 𝜋 ( 𝑎 𝑏 ) 2

Q17:

Find, in terms of 𝜋 , the volume of the region that lies within both the sphere with equation 𝑥 + 𝑦 + 𝑧 = 4 2 2 2 and the cylinder with equation 𝑥 + 𝑦 = 1 2 2 .

  • A 𝜋 3 8 − 3 3 2
  • B 2 √ 3 𝜋
  • C 2 𝜋 3 8 − 3 3 2
  • D 4 𝜋 3 8 − 3 3 2
  • E √ 3 𝜋

Q18:

Find the volume inside both the sphere 𝑥 + 𝑦 + 𝑧 = 1 2 2 2 and the cone 𝑧 = √ 𝑥 + 𝑦 2 2 .

  • A 𝜋 3 1 − 1 √ 2
  • B 𝜋 1 2 7 − 3 √ 3
  • C 𝜋 9 1 − 1 √ 2
  • D 2 𝜋 3 1 − 1 √ 2
  • E 𝜋 6 1 − 1 √ 2

Q19:

Find the volume 𝑉 of the solid bounded by the three coordinate planes and the plane 3 𝑥 + 2 𝑦 + 5 𝑧 = 6 .

Q20:

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 2 has at least one real solution for 𝑥 ? Rounding the value to four decimal places.

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