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 360∘ about the π‘₯-axis. The region is bounded by the curve 𝑦=12π‘₯+6, the line π‘₯=0, the line π‘₯=1, and the π‘₯-axis. Determine the position of the center of mass of that solid using integration.

  • Aο€Ό6547,0
  • Bο€Ό8594,0
  • Cο€Ό3547,0
  • Dο€Ό6594,0
  • Eο€Ό6394,0

Q2:

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

  • Aο€Ό34,38
  • Bο€Ό910,34
  • Cο€Ό38,910
  • Dο€Ό34,910
  • Eο€Ό38,95

Q3:

A solid is formed by rotating the finite region bounded by the curve π‘¦βˆ’3𝑦=π‘₯ and the 𝑦-axis through 360∘ about the 𝑦-axis. Determine the coordinates of the center of mass of that solid using symmetry.

  • A(1.5,0)
  • B(βˆ’1.5,0)
  • C(0,1.5)
  • D(0,βˆ’1.5)
  • E(0,βˆ’3)

Q4:

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

  • Aο€Ό125,18
  • Bο€Ό15,92
  • C(15,18)
  • Dο€Ό125,9
  • Eο€Ό125,92

Q5:

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

  • Aο€Ό415,47
  • Bο€Ό815,835
  • Cο€Ό47,815
  • Dο€Ό415,835
  • Eο€Ό815,47

Q6:

A solid of revolution is formed by rotating the finite region bounded by the curve 𝑦+(π‘₯+5)=9 through 180∘ about the π‘₯-axis. Using symmetry, determine the coordinates of its center of mass.

  • A(βˆ’2.5,0)
  • B(βˆ’5,0)
  • C(0,5)
  • D(5,0)
  • E(0,βˆ’5)

Q7:

The region bounded by the curve 𝑦=5π‘₯sin, the lines π‘₯=5πœ‹2 and π‘₯=9πœ‹2, and the π‘₯-axis is rotated 360∘ about the π‘₯-axis, forming a solid of revolution. Find the π‘₯-coordinate of the solid’s center of mass.

  • Aπœ‹
  • B7πœ‹4
  • C7πœ‹2
  • D2πœ‹
  • E7πœ‹

Q8:

The region bounded by the curve 𝑦=4π‘₯, the lines π‘₯=5 and π‘₯=9, and the π‘₯-axis is rotated by 360∘ about the π‘₯-axis, forming a solid of revolution. Giving your answer to one decimal place, find the π‘₯ -coordinate of the solid's center of mass.

Q9:

A uniform lamina occupies the finite region bounded by the curve 𝑦=16π‘Žπ‘₯ and the line π‘₯=π‘Ž, where π‘Ž is a positive constant. Using integration, find the center of mass of the lamina.

  • Aο€Ό3π‘Ž20,0
  • Bο€Ό3π‘Ž5,0
  • Cο€Ό3π‘Ž10,0
  • Dο€Ύ3π‘Ž10,3π‘Ž4
  • Eο€Ύ3π‘Ž5,3π‘Ž4

Q10:

The region bounded by the curve 𝑦=10√π‘₯3, the line π‘₯=72, and the π‘₯-axis is rotated through 360∘ about the π‘₯-axis. Determine the center of mass of the solid formed.

  • Aο€Ώ6√1425,5√148
  • Bο€Ώ6√1425,0
  • Cο€Ό74,0
  • Dο€Ό73,0
  • Eο€Ώ73,5√148

Q11:

A uniform lamina is bounded by the curve 𝑦=4π‘₯sin, where 0≀π‘₯β‰€πœ‹, and the line 𝑦=0. Find, by integration, its center of mass.

  • Aο€»πœ‹2,2πœ‹ο‡
  • Bο€»πœ‹4,πœ‹2
  • Cο€»πœ‹2,πœ‹2
  • Dο€»πœ‹2,πœ‹4
  • Eο€»πœ‹2,πœ‹ο‡

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