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Lesson: Triple Integrals

Worksheet • 20 Questions

Q1:

Evaluate the triple integral ο„Έ ο„Έ ο„Έ π‘₯ 𝑦 𝑧 π‘₯ 𝑦 𝑧 3 0 2 0 1 0 d d d .

Q2:

Evaluate the triple integral ο„Έ ο„Έ ο„Έ 𝑧 π‘₯ π‘₯ 𝑦 𝑧 1 0 𝑧 0 𝑦 0 2 d d d .

  • A 1 7 2
  • B 1 1 2
  • C 1 2
  • D 1 3
  • E1

Q3:

Find the centre of mass of the solid 𝑆 = { ( π‘₯ , 𝑦 , 𝑧 ) 0 ≀ π‘₯ ≀ 1 , 0 ≀ 𝑦 ≀ 1 , 0 ≀ 𝑧 ≀ 1 βˆ’ π‘₯ βˆ’ 𝑦 } : with the given density function 𝜌 ( π‘₯ , 𝑦 , 𝑧 ) = 1 .

  • A ο€Ό 1 4 , 1 4 , 1 4 
  • B ( 4 , 4 , 4 )
  • C ο€Ό 1 8 , 1 8 , 1 8 
  • D ο€Ό 1 6 , 1 6 , 1 6 
  • E ο€Ό 1 2 , 1 2 , 1 2 

Q4:

Evaluate the triple integral ο„Έ ο„Έ ο„Έ 1 𝑧 𝑦 π‘₯ 1 0 1 βˆ’ π‘₯ 0 1 βˆ’ π‘₯ βˆ’ 𝑦 0 d d d .

  • A 1 6
  • B 1 4
  • C βˆ’ 1 2
  • D 1 2
  • E 1 3

Q5:

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

Q6:

Find the volume inside the cone 𝑧 = √ π‘₯ + 𝑦 2 2 , where 0 ≀ 𝑧 ≀ 3 .

  • A 9 πœ‹
  • B 9 2
  • C 1 8 πœ‹
  • D √ 3 πœ‹
  • E ο€» 9 βˆ’ 2 √ 3  πœ‹

Q7:

Find the centre of mass of the solid 𝑆 =  ( π‘₯ , 𝑦 , 𝑧 ) π‘₯ β‰₯ 0 , 𝑦 β‰₯ 0 , 𝑧 β‰₯ 0 , π‘₯ + 𝑦 + 𝑧 ≀ π‘Ž  : 2 2 2 2 with the given density function 𝜌 ( π‘₯ , 𝑦 , 𝑧 ) = 1 .

  • A ο€Ό 3 π‘Ž 8 , 3 π‘Ž 8 , 3 π‘Ž 8 
  • B ο€Ό 8 3 π‘Ž , 8 3 π‘Ž , 8 3 π‘Ž 
  • C ο€Ό 3 π‘Ž 1 6 , 3 π‘Ž 1 6 , 3 π‘Ž 1 6 
  • D ο€» π‘Ž 2 , π‘Ž 2 , π‘Ž 2 
  • E ο€Ό 2 π‘Ž 3 , 2 π‘Ž 3 , 2 π‘Ž 3 

Q8:

Evaluate the triple integral ο„Έ ο„Έ ο„Έ 𝑧 π‘₯ π‘₯ 𝑧 𝑦 𝑒 1 𝑦 0 0 2 1 𝑦 d d d .

  • A 1 6
  • B βˆ’ 5 6
  • C 1 2
  • D 1 3
  • E1

Q9:

Evaluate the triple integral ο„Έ ο„Έ ο„Έ 𝑧 𝑒 π‘₯ 𝑦 𝑧 1 0 𝑧 0 𝑦 0 𝑦 2 d d d .

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

Q10:

Evaluate the triple integral ο„Έ ο„Έ ο„Έ 1 π‘₯ 𝑦 𝑧 2 1 4 2 3 0 d d d .

Q11:

Find the centre of mass of the solid 𝑆 = { ( π‘₯ , 𝑦 , 𝑧 ) 0 ≀ π‘₯ ≀ 1 , 0 ≀ 𝑦 ≀ 1 , 0 ≀ 𝑧 ≀ 1 } : with the given density function 𝜌 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ 𝑦 𝑧 .

  • A ο€Ό 2 3 , 2 3 , 2 3 
  • B ο€Ό 3 2 , 3 2 , 3 2 
  • C ο€Ό 1 3 , 1 3 , 1 3 
  • D ο€Ό 1 2 , 1 2 , 1 2 
  • E ( 1 , 1 , 1 )

Q12:

Find the volume inside the cone 𝑧 = π‘₯ + 𝑦 2 2 , where 0 ≀ 𝑧 ≀ 4 .

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

Q13:

Find the volume inside the elliptic cylinder π‘₯ π‘Ž + 𝑦 𝑏 = 1 2 2 2 2 , where 0 ≀ 𝑧 ≀ 2 .

  • A 2 πœ‹ π‘Ž 𝑏
  • B 4 πœ‹ ( π‘Ž 𝑏 ) 2
  • C 4 πœ‹ π‘Ž 𝑏
  • D πœ‹ ( π‘Ž 𝑏 ) 2
  • E πœ‹ π‘Ž 𝑏

Q14:

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

  • A 4 πœ‹ 3 ο€½ 8 βˆ’ 3  3 2
  • B √ 3 πœ‹
  • C 2 √ 3 πœ‹
  • D 2 πœ‹ 3 ο€½ 8 βˆ’ 3  3 2
  • E πœ‹ 3 ο€½ 8 βˆ’ 3  3 2

Q15:

Find the volume inside both the sphere π‘₯ + 𝑦 + 𝑧 = 1 2 2 2 and the cone 𝑧 = √ π‘₯ + 𝑦 2 2 .

  • A 2 πœ‹ 3 ο€Ώ 1 βˆ’ 1 √ 2 
  • B πœ‹ 6 ο€Ώ 1 βˆ’ 1 √ 2 
  • C πœ‹ 1 2 ο€» 7 βˆ’ 3 √ 3 
  • D πœ‹ 9 ο€Ώ 1 βˆ’ 1 √ 2 
  • E πœ‹ 3 ο€Ώ 1 βˆ’ 1 √ 2 

Q16:

Find the centre of mass of the solid 𝑆 = { ( π‘₯ , 𝑦 , 𝑧 ) 0 ≀ π‘₯ ≀ 1 , 0 ≀ 𝑦 ≀ 1 , 0 ≀ 𝑧 ≀ 1 } : with the given density function 𝜌 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ + 𝑦 + 𝑧 2 2 2 .

  • A ο€Ό 7 1 2 , 7 1 2 , 7 1 2 
  • B ο€Ό 5 7 , 5 7 , 5 7 
  • C ( 1 , 1 , 1 )
  • D ο€Ό 5 1 2 , 5 1 2 , 5 1 2 
  • E ο€Ό 1 2 7 , 1 2 7 , 1 2 7 

Q17:

Let π‘Ž , 𝑏 , and 𝑐 be real numbers selected randomly from the interval ] 0 , 1 [ . What is the probability that the equation π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐 = 0 2 has at least one real solution for π‘₯ ? Rounding the value to four decimal places.

Q18:

Evaluate the triple integral ο„Έ ο„Έ ο„Έ π‘₯ 𝑦 𝑧 𝑧 𝑦 π‘₯ 1 0 π‘₯ 0 𝑦 0 d d d .

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

Q19:

Find the centre of mass of the solid 𝑆 =  ( π‘₯ , 𝑦 , 𝑧 ) 𝑧 β‰₯ 0 , π‘₯ + 𝑦 + 𝑧 ≀ π‘Ž  : 2 2 2 2 with the given density function 𝜌 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ + 𝑦 + 𝑧 2 2 2 .

  • A ο€Ό 0 , 0 , 5 π‘Ž 1 2 
  • B ο€Ό 0 , 0 , 1 2 5 π‘Ž 
  • C ο€Ό 0 , 0 , 5 π‘Ž 6 
  • D ο€Ό 0 , 0 , 5 π‘Ž 8 
  • E ο€Ό 0 , 0 , 5 π‘Ž 3 

Q20:

Evaluate the triple integral ο„Έ ο„Έ ο„Έ 𝑦 𝑧 π‘₯ 𝑧 𝑦 2 1 𝑦 0 𝑧 0 2 2 d d d .

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