Lesson Worksheet: Center of Mass and Integration Mathematics

A solid is formed when a finite region is rotated through about the -axis. The region is bounded by the curve , the line , the line , and the -axis. Determine the position of the center of mass of that solid using integration.

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

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 center of mass of that solid using symmetry.

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

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

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 center of mass.

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 solid’s center of mass.

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 center of mass.

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

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