# 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 about -axis. The region is bounded by the curve , the line , the line , and the -axis. Determine the position of the centre of mass of that solid using integration.

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**Q2: **

Using integration, determine the centre of mass of a uniform lamina that occupies the finite region bounded by the curve , the -axis, and the line .

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**Q3: **

A solid is formed by rotating the finite region bounded by the curve and the -axis through about the -axis. Determine the coordinates of the centre of mass of that solid using symmetry.

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**Q4: **

A uniform lamina is bounded by the curve , the -axis, and the line . Find, by integration, its centre of mass.

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**Q5: **

Find the centre of mass of the uniform lamina bounded by the curve and the straight line , as shown in the figure.

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**Q6: **

A solid of revolution is formed by rotating the finite region bounded by the curve through about the -axis. Using symmetry, determine the coordinates of its centre of mass.

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**Q7: **

The region bounded by the curve , the lines and , and the -axis is rotated about the -axis, forming a solid of revolution. Find the -coordinate of the solidβs centre of mass.

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**Q8: **

The region bounded by the curve , the lines and , and the -axis is rotated by about the -axis, forming a solid of revolution. Giving your answer to one decimal place, find the -coordinate of the solid's centre of mass.

**Q9: **

A uniform lamina occupies the finite region bounded by the curve and the line , where is a positive constant. Using integration, find the centre of mass of the lamina.

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**Q10: **

The region bounded by the curve , the line , and the -axis is rotated through about the -axis. Determine the centre of mass of the solid formed.

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**Q11: **

A uniform lamina is bounded by the curve , where , and the line . Find, by integration, its centre of mass.

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