Worksheet: Center of Mass and Integration

In this worksheet, we will practice using integration to find the centre of mass of a region bounded by a curve of a function.

Q1:

A solid is formed when a finite region is rotated through 3 6 0 ∘ about π‘₯ -axis. The region is bounded by the curve 𝑦 = 1 2 π‘₯ + 6  , the line π‘₯ = 0 , the line π‘₯ = 1 , and the π‘₯ -axis. Determine the position of the centre of mass of that solid using integration.

  • A ο€Ό 6 5 9 4 , 0 
  • B ο€Ό 3 5 4 7 , 0 
  • C ο€Ό 6 3 9 4 , 0 
  • D ο€Ό 6 5 4 7 , 0 
  • E ο€Ό 8 5 9 4 , 0 

Q2:

Using integration, determine the centre of mass of a uniform lamina that occupies the finite region bounded by the curve 𝑦 = 3 π‘₯  , the π‘₯ -axis, and the line π‘₯ = 1 .

  • A ο€Ό 3 4 , 3 8 
  • B ο€Ό 3 8 , 9 5 
  • C ο€Ό 3 4 , 9 1 0 
  • D ο€Ό 9 1 0 , 3 4 
  • E ο€Ό 3 8 , 9 1 0 

Q3:

A solid is formed by rotating the finite region bounded by the curve 𝑦 βˆ’ 3 𝑦 = π‘₯  and the 𝑦 -axis through 3 6 0 ∘ about the 𝑦 -axis. Determine the coordinates of the centre of mass of that solid using symmetry.

  • A ( 0 , 1 . 5 )
  • B ( 1 . 5 , 0 )
  • C ( βˆ’ 1 . 5 , 0 )
  • D ( 0 , βˆ’ 3 )
  • E ( 0 , βˆ’ 1 . 5 )

Q4:

A uniform lamina is bounded by the curve 𝑦 = 6 √ π‘₯ , the π‘₯ -axis, and the line π‘₯ = 4 . Find, by integration, its centre of mass.

  • A ο€Ό 1 2 5 , 1 8 
  • B ο€Ό 1 2 5 , 9 2 
  • C ο€Ό 1 5 , 9 2 
  • D ( 1 5 , 1 8 )
  • E ο€Ό 1 2 5 , 9 

Q5:

Find the centre of mass of the uniform lamina bounded by the curve 𝑦 = 3 π‘₯ 2  and the straight line 𝑦 = 3 π‘₯ 2 , as shown in the figure.

  • A ο€Ό 4 7 , 8 1 5 
  • B ο€Ό 4 1 5 , 8 3 5 
  • C ο€Ό 8 1 5 , 4 7 
  • D ο€Ό 8 1 5 , 8 3 5 
  • E ο€Ό 4 1 5 , 4 7 

Q6:

A solid of revolution is formed by rotating the finite region bounded by the curve 𝑦 + ( π‘₯ + 5 ) = 9   through 1 8 0 ∘ about the π‘₯ -axis. Using symmetry, determine the coordinates of its centre of mass.

  • A ( 5 , 0 )
  • B ( βˆ’ 2 . 5 , 0 )
  • C ( 0 , 5 )
  • D ( βˆ’ 5 , 0 )
  • E ( 0 , βˆ’ 5 )

Q7:

The region bounded by the curve 𝑦 = 5 π‘₯ s i n , the lines π‘₯ = 5 πœ‹ 2 and π‘₯ = 9 πœ‹ 2 , and the π‘₯ -axis is rotated 3 6 0 ∘ about the π‘₯ -axis, forming a solid of revolution. Find the π‘₯ -coordinate of the solid’s centre of mass.

  • A 2 πœ‹
  • B πœ‹
  • C 7 πœ‹
  • D 7 πœ‹ 2
  • E 7 πœ‹ 4

Q8:

The region bounded by the curve 𝑦 = 4 π‘₯ , the lines π‘₯ = 5 and π‘₯ = 9 , and the π‘₯ -axis is rotated by 3 6 0 ∘ about the π‘₯ -axis, forming a solid of revolution. Giving your answer to one decimal place, find the π‘₯ -coordinate of the solid's centre of mass.

Q9:

A uniform lamina occupies the finite region bounded by the curve 𝑦 = 1 6 π‘Ž π‘₯  and the line π‘₯ = π‘Ž , where π‘Ž is a positive constant. Using integration, find the centre of mass of the lamina.

  • A ο€Ύ 3 π‘Ž 1 0 , 3 π‘Ž 4  
  • B ο€Ό 3 π‘Ž 1 0 , 0 
  • C ο€Ύ 3 π‘Ž 5 , 3 π‘Ž 4  
  • D ο€Ό 3 π‘Ž 5 , 0 
  • E ο€Ό 3 π‘Ž 2 0 , 0 

Q10:

The region bounded by the curve 𝑦 = 1 0 √ π‘₯ 3 , the line π‘₯ = 7 2 , and the π‘₯ -axis is rotated through 3 6 0 ∘ about the π‘₯ -axis. Determine the centre of mass of the solid formed.

  • A ο€Ό 7 4 , 0 
  • B ο€Ώ 6 √ 1 4 2 5 , 0 
  • C ο€Ό 7 3 , 0 
  • D ο€Ώ 6 √ 1 4 2 5 , 5 √ 1 4 8 
  • E ο€Ώ 7 3 , 5 √ 1 4 8 

Q11:

A uniform lamina is bounded by the curve 𝑦 = 4 π‘₯ s i n , where 0 ≀ π‘₯ ≀ πœ‹ , and the line 𝑦 = 0 . Find, by integration, its centre of mass.

  • A ο€» πœ‹ 2 , 2 πœ‹ 
  • B ο€» πœ‹ 2 , πœ‹ 2 
  • C ο€» πœ‹ 2 , πœ‹ 
  • D ο€» πœ‹ 4 , πœ‹ 2 
  • E ο€» πœ‹ 2 , πœ‹ 4 

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